Nephi not only made it very clear that there would be only one Messiah, he also revealed the name of the Messiah—"Jesus Christ, the Son of God"—and even gave a timeframe when this one Messiah would be born. Nephi did not pin the date down precisely, but he did say that it would be "six hundred years from the time that [Lehi and his family] left Jerusalem." In other words, Christ’s birth would be close enough to 600 years that people would have recognized that Jesus was the Messiah, and would have been convinced that there was no need to look for another.
Book of Mormon Central, "How Does Prophecy Shape the Book of Mormon’s Content and Structure? (Words of Mormon 1:4)," KnoWhy 498 (January 15, 2019).