Do you know of a prophet in the Old Testament between Moses and Samuel?
Joshua is the only one from whom we have much of a prophetic record, and in that record he seems to pretty much limit himself to what Moses taught. So Jesus, having explained that Moses testified about His coming and His teaching, moves on to Samuel, I would guess.
Elsewhere, such as in Jacob 7: 11, we are told that everyone from Adam down prophesied about Jesus. You can read some of those prophecies in the Pearl of Great Price.
If you are wondering, a significant part of the early Book of Mormon narrative concerns Lehi obtaining an early copy of what would be close to the Hebrew scriptures extant ca. 600 BC. So we see a bit of discussion of Moses and Isaiah and others of the Old Testament prophets in the Book of Mormon.
It is interesting that Jesus chose to teach a principle that Peter taught (See Acts 3: 22 - 26.), and that Joseph Smith's translation of Jesus' words here seems to follow Peter's words really closely.
Some people worry about this, thinking it must be evidence of plagiarism or something. If we assume that Joseph Smith was a charlatan, we might think so.
On the other hand, God doesn't seem to worry about His prophets plagiarizing each other. Prophets quote each other quite liberally.
So it can really only confirm the assumptions you start with, for Joseph Smith to have Jesus apparently quoting Peter. He was teaching the same principle Peter was teaching in Jerusalem about the same time.
For the reasons it so closely parallels the King James version of the Bible, Joseph Smith's formal education was limited to three years of what we now call elementary school. Beyond that, the Bible formed the basis of the bulk of his education in English and philosophy, at minimum.
Having done some translation myself, between English and Japanese, I know that, when I go out of my way to use vocabulary and phrasing I'm not familiar with, not only is it hard, but I tend to produce things that don't match existing idiom -- they sound strange, and they tend not to make much sense.
So I find myself both consciously and subconsciously resorting to idioms I have heard, and that means I often quote scriptures near verbatim when what I am translating is material that is scriptural and it happens to be scriptures that I remember.
For these reasons, among others, it does not surprise me that the Book of Mormon quotations from the Bible so closely parallel the KJT.
PyRulez brings it to my attention that I let the overall focus of my response shift too much.
Here is a discussion on the Church website of the concept that all the prophets from the beginning have testified about Jesus Christ's coming and the work He would do:
From that article, I would particularly note 1st Nephi chs. 8, 11-14, the prophetic dreams which Lehi and Nephi saw.
Also, Samuel the Lamanite, mentioned by PyRulez and others, is recorded in Helaman 8, ca. verses 16 and 18, to have talked about a number of prophets from Abraham down who prophesied of Jesus and of the redemption. If this is the Samuel that the Savior was mentioning, per your question and JBH's and PyRulz's comments, I would say that it is not just because Samuel taught the concept that all the prophets testified of Him, but because Samuel testified in great detail about the Savior's coming. Very definitely worth reading.
And, as I mentioned, many verses in the Pearl of Great Price, esp. the Book of Moses, two of which may be of particular interest:
Moses being told his ministry would symbolize the ministry of Jesus in Moses 1: 6.
And Enoch prophesying of Jesus Christ before the flood in (among other places) Moses 6: 51, 52. In this particular scripture, Enoch is explaining that Adam himself was taught, not only of the coming of a Savior who would redeem men from their sins, but of the name of the Savior.