During the Millennial reign of Jesus Christ, missionary work will continue to spread the gospel. The message will reach to many more than it had prior to the Second Coming, ’for every ear shall hear it, and every knee shall bow, and every tongue shall confess’ (DC 88:104). The Lord will also fill the world with knowledge long since hidden:
’Yea, verily I say unto you, in that day when the Lord shall come, he shall reveal all things—
Things which have passed, and hidden things which no man knew, things of the earth, by which it was made, and the purpose and the end thereof—
Things most precious, things that are above, and things that are beneath, things that are in the earth, and upon the earth, and in heaven’ (DC 101:32-34).
Not only will the knowledge of the gospel be spread during the Millennium, but the number of people with a personal knowledge of their Lord and Savior, Jesus Christ, will increase. In other words, those who had exhibited great faith in their Lord but were not privileged to see His face in mortality will receive that privilege during the Millennium. Therefore, their faith will be dormant in that thing (Alma 32:34) for it will be replaced with the ‘knowledge of the Lord’ (Ether 3:19).
Joseph Fielding Smith
"The gospel will be taught far more intensely and with greater power during the millennium until all the inhabitants of the earth shall embrace it. Satan shall be bound so that he cannot tempt any man. Should any man refuse to repent and accept the gospel under those conditions then he would be accursed. Through the revelations given to the prophets we learn that during the reign of Jesus Christ for a thousand years, eventually all people will embrace the truth. Isaiah prophesied of the millennium as follows: [Isa 11:6-9 quoted]
“This chapter in Isaiah Moroni quoted to the Prophet Joseph Smith and said to him it was about to be fulfilled. If the knowledge of the Lord covers the earth as the waters do the sea, then it must be universally received. Moreover, the promise of the Lord through Jeremiah is that it will no longer be necessary for anyone to teach his neighbor, ’… saying, Know the Lord: for they shall all know me, from the least of them unto the greatest of them, saith the Lord.’” (Answers to Gospel Questions, vol. 1, p. 111)
"Some members of the Church have an erroneous idea that when the millennium comes all of the people are going to be swept off the earth except righteous members of the Church. That is not so. There will be millions of people, Catholics, Protestants, agnostics, Mohammedans, people of all classes, and of all beliefs, still permitted to remain upon the face of the earth, but they will be those who have lived clean lives, those who have been free from wickedness and corruption. All who belong, by virtue of their good lives, to the terrestrial order, as well as those who have kept the celestial law, will remain upon the face of the earth during the millennium.
Millenniuma-Smith, Joseph FieldingTP
“Eventually, however, the knowledge of the Lord will cover the earth as the waters do the sea. But there will be need for the preaching of the gospel, after the millennium is brought in, until all men are either converted or pass away. In the course of the thousand years all men will either come into the Church, or kingdom of God, or they will die and pass away.” (Doctrines of Salvation, 1:86 as taken from Commentaries on Isaiah in the Book of Mormon, ed. by K. Douglas Bassett, [American Fork, UT: Covenant Publishing Co., 2003], 181-182)