What does Mormonism have to offer in regards to salvation through Jesus Christ that was not already taught in the New Testament?

From what little I understand, Mormonism offers one thing regarding salvation through Jesus Christ that is not only different from Biblical teaching but is diametrically opposed.

The Bible teaches that God the Father is eternally, infinitely unchangeable; that He always has been and always will be the same...from everlasting to everlasting. He did not evolve and He did not become...He is.

LDS belief is that God was once a man. Fifth LDS President Lorenzo Snow made famous a catchy couplet that goes like this: "“As man now is, God once was; as God now is, man may be”.

An article by Gerald N. Lund, Teacher Support Consultant for the Church Education System, on the Church of Jesus Christ website has this to say about the couplet:

To my knowledge there has been no “official” pronouncement by the First Presidency declaring that President Snow’s couplet is to be accepted as doctrine. But that is not a valid criteria for determining whether or not it is doctrine. Generally, the First Presidency issues official doctrinal declarations when there is a general misunderstanding of the doctrine on the part of many people. Therefore, the Church teaches many principles which are accepted as doctrines but which the First Presidency has seen no need to declare in an official pronouncement. This particular doctrine has been taught not only by Lorenzo Snow, fifth President of the Church, but also by others of the Brethren before and since that time.

Indeed, the Prophet Joseph Smith is reported to have corroborated this teaching and taught it himself.

President Snow’s son LeRoi later told that the Prophet Joseph Smith confirmed the validity of the revelation Elder Snow had received: “Soon after his return from England, in January, 1843, Lorenzo Snow related to the Prophet Joseph Smith his experience in Elder Sherwood’s home. This was in a confidential interview in Nauvoo. The Prophet’s reply was: ‘Brother Snow, that is a true gospel doctrine, and it is a revelation from God to you.’” (LeRoi C. Snow, Improvement Era, June 1919, p. 656.)

The Prophet Joseph Smith himself publicly taught the doctrine the following year, 1844, during a funeral sermon of Elder King Follett: “God himself was once as we are now, and is an exalted man, and sits enthroned in yonder heavens! … It is the first principle of the Gospel to know for a certainty the Character of God, and to know that we may converse with him as one man converses with another, and that he was once a man like us; yea, that God himself, the Father of us all, dwelt on an earth, the same as Jesus Christ himself did.

That God was once as we are now and that God is currently an exalted man is declared by the Prophet Joseph Smith to be the first principle of the Gospel.

The biblical hope of salvation is to live in God's presence forever.

The Mormon hope of salvation is to become Gods in our own right "as God now is, man may be"

This cannot be considered a further, corrective revelation of what the bible has lost clarity on since it is a direct contradiction of consistently clear Scripture. It may indeed be "another" revelation but, as such, it is outside the scope of biblical Christianity.

An aside - If it can be assumed that the creation account in Genesis chapter one takes place after God the Father became God (what non-God could do such things), one wonders upon what earth did the man who was to become that God live prior to anything having been created.

Mormonism teaches that salvation comes through Jesus Christ, like any Christian faith.

What does Mormonism have to offer in regards to salvation?

The whole truth which includes priesthood authority to perform ordinances necessary for salvation.

Ephesians 4:5

5 One Lord, one faith, one baptism,

As OP pointed out billions have become Christian in the last 2,000 years, but of varying denominations¹. To me this indicates confusion or disagreements over doctrine contained in the Bible; if all agreed all would belong to the same denomination. The Church of Jesus Christ of Latter-day Saints believe in the Bible but also the Book of Mormon, they support each other. An analogy I've heard to understand this is if you take one point (Bible/testament) you can draw multiple lines(understanding/denominations) through it, but two points(other scripture/multiple testaments) you can only get one line through.²

Why do we need "Another Testament of Jesus Christ" when the first testament has proven sufficient to save one's soul?

The Bible, or other scripture, are not sufficient to save one's soul. I don't believe most denominations believe this to be the case. They are a guide or instruction manual. If you are missing guidance, instructions, or have confusing instructions you will not be able to find the whole truth, which is needed to lead one to salvation.

The gospel is fixed, not to be revised or have somebody else with another "gospel/testament" show up

The Church of Jesus Christ do not believe the Book of Mormon contains a different gospel. From Gospel Topics manual:

The Book of Mormon is another witness of Jesus Christ and confirms the truths found in the Holy Bible. Far from undermining the Bible, the Book of Mormon supports its testimony of Jesus Christ. One passage says that the Book of Mormon “shall establish the truth” of the Bible “and shall make known to all kindreds, tongues, and people, that the Lamb of God is the Son of the Eternal Father, and the Savior of the world; and that all men must come unto him, or they cannot be saved” (1 Nephi 13:40).

Rhetorical/Hypothetical questions:

• If one or more additional accounts of Jesus' life, ministry, and/or teachings were found written by someone else (like apocrypha or one of the apostle's account not contained in the bible) and it was confirmed to be 100% legitimate and multiple sources confirmed the contents (maybe it was from an attendee at the sermon on the mount), would this be considered scripture?
• When the 2nd coming happens and Jesus comes again and says something not found in the Bible is this considered scripture?

_{1 How many Christian denominations are there?}

_{2 Why is the Book of Mormon Essential?}

What does Mormonism have to offer in regards to salvation that Jesus Christ had not already accomplished?

In some sense, nothing. The atonement of Jesus Christ is what saves us. Faith in Jesus Christ is what saves us. The scriptures, however many they may be, can only lead us to Christ but never save. An illiterate person can be saved. But in this sense you have to question what any denomination has to offer, or what the visible church had to offer that made it a good idea to establish one in the New Testament. Or even what 98% of the Bible "has to offer".

However, Jesus has also said that we need to be baptized (John 3:5). In his time, he gave his apostles authority to baptize in his name. Without this authority, a baptism is just going through the motions. We believe this authority was lost early in the church history and restored by angels who conferred this authority to Joseph Smith. So in that sense, the true church should have authority to perform ordinances that Christ described as necessary for salvation.

The Church of Jesus Christ of Latter-Day Saints claims to be a restoration of the original church Christ founded. Literally the church of Jesus Christ. Whatever it was that having a church in the first place offered, is what "Mormonism offers".

So why do we need "Another Testament of Jesus Christ" when the first testament, i.e the Bible, (specifically the New Testament) has proven sufficient to save one's soul?

Moses was given a stick that turned into a snake to prove he was sent by God. While the Book of Mormon does restore or clarify parts of the doctrine, again, it's not the doctrine that saves, any amount of it. Think of the Book of Mormon as Joseph Smith's stick that turns into a snake.

The title changed so a significant portion of my answer doesn't really fit the title anymore. To answer the changed title, here are some of the things the Book of Mormon teaches with regards to salvation, that are disputed among the different denominations. So, in that sense, these are not "new" but really an additional witness that clarifies points and gets rid of confusion and misinterpretation. Note please the context. Joseph, in his First Vision, was told not to join any of the denominations of his day, that they were all wrong. He wasn't told "this point of their doctrine is wrong, and instead this is right". So, when he came out of that, he knew God and Jesus existed, but apart from that not really much more. The Bible is still right, but still no idea how to interpret it. He possibly knew less than what he thought he knew before. So for him when the time came to translate the Book of Mormon, the additional witness the Book of Mormon provided was very important to figure out what the right doctrine is.

Freedom of choice (as opposed to Calvinist beliefs): 2 Nephi 2: 26-27

26 And the Messiah cometh in the fulness of time, that he may redeem the children of men from the fall. And because that they are redeemed from the fall they have become free forever, knowing good from evil; to act for themselves and not to be acted upon, save it be by the punishment of the law at the great and last day, according to the commandments which God hath given.

27 Wherefore, men are free according to the flesh; and all things are given them which are expedient unto man. And they are free to choose liberty and eternal life, through the great Mediator of all men, or to choose captivity and death, according to the captivity and power of the devil; for he seeketh that all men might be miserable like unto himself.

Baptism is a commandment, and essential 3 Nephi 27:16 (among others)

16 And it shall come to pass, that whoso repenteth and is baptized in my name shall be filled; and if he endureth to the end, behold, him will I hold guiltless before my Father at that day when I shall stand to judge the world.

... except for children (as opposed to child baptism practices) Moroni 8:8

8 Listen to the words of Christ, your Redeemer, your Lord and your God. Behold, I came into the world not to call the righteous but sinners to repentance; the whole need no physician, but they that are sick; wherefore, little children are whole, for they are not capable of committing sin; wherefore the curse of Adam is taken from them in me, that it hath no power over them; and the law of circumcision is done away in me.

Priesthood authority is necessary for baptism, and is conferred (as opposed to priesthood of all believers) 3 Nephi 11:21-22

21 And the Lord said unto him: I give unto you power that ye shall baptize this people when I am again ascended into heaven.

22 And again the Lord called others, and said unto them likewise; and he gave unto them power to baptize. And he said unto them: On this wise shall ye baptize; and there shall be no disputations among you.

Christ saves us from our sins, not in our sins Helaman 5:10-11

10 And remember also the words which Amulek spake unto Zeezrom, in the city of Ammonihah; for he said unto him that the Lord surely should come to redeem his people, but that he should not come to redeem them in their sins, but to redeem them from their sins.

11 And he hath power given unto him from the Father to redeem them from their sins because of repentance; therefore he hath sent his angels to declare the tidings of the conditions of repentance, which bringeth unto the power of the Redeemer, unto the salvation of their souls.

• Resurrection is unconditional (not sure who teaches otherwise, but I am sure some denomination is going to teach only the righteous are resurrected. Maybe JW)
• There is a time between death and resurrection where spirits await their resurrection (as opposed to soul sleep/ mortalism)
• the second death suffered by the wicked is being cast out from Gods presence, not a literal destruction of their immortal self

Alma 40

I wouldn't call any of these points new (after all, "Christ suffered for our sins so we can repent and return to God" is so simple I can literally write it in one sentence), but you have to admit these are points that are contentious among denominations and as such can't be considered to be clear beyond doubt in the Bible.