You are not seeing what Jesus said though.
"And his disciples asked him, saying, Why then say the scribes that Elias must first come? And Jesus answered and said unto them, Elias truly shall first come, and restore all things. But I say unto you, That Elias is come already, and they knew him not, but have done unto him whatsoever they listed. Likewise shall also the Son of man suffer of them. Then the disciples understood that he spake unto them of John the Baptist."
They only understood it as John the Baptist. Not that Elijah is John the Baptist.
The coming of Elijah to restore all things is still future. Then Jesus said, "but".
Is Jesus contradicting Himself then? No.
Jesus was pointing out that John the Baptist in the spirit of Elijah already came. John told every one he was not Elijah. Was John lying as well? John 1:20-22
"And he confessed, and denied not; but confessed, I am not the Christ. And they asked him, What then? Art thou Elias? And he saith, I am not. Art thou that prophet? And he answered, No. Then said they unto him, Who art thou? that we may give an answer to them that sent us. What sayest thou of thyself?"
If Jesus was correct even John the Baptist did not know he was Elias. Which means reincarnation will not even work as it is posited in this thread, as a means of escape from sheol. No one even knows who they are. How can they consciously be redeemed?
Luke explains who John was in Luke 1.
"But the angel said unto him, Fear not, Zacharias: for thy prayer is heard; and thy wife Elisabeth shall bear thee a son, and thou shalt call his name John. And thou shalt have joy and gladness; and many shall rejoice at his birth. For he shall be great in the sight of the Lord, and shall drink neither wine nor strong drink; and he shall be filled with the Holy Ghost, even from his mother's womb. And many of the children of Israel shall he turn to the Lord their God. And he shall go before him in the spirit and power of Elias, to turn the hearts of the fathers to the children, and the disobedient to the wisdom of the just; to make ready a people prepared for the Lord."
Jesus was not teaching at the moment, and certainly not reincarnation. Jesus could have been saying that an Elias would come, but not the reincarnation of Elias. The prophecy was fulfilled in John the Baptist with the spirit and power of Elijah, not Elijah himself. The way Jesus put it, Elijah could still come as prophecied. But in preparation for the first coming, John the Baptist, a totally different soul, was the one who came to prepare the way of the Messiah.
Jesus was not saying John the Baptist was Elijah, but that John the Baptist fulfulled the prophecy in part, not Elijah reincarnated. If Elijah came back it would not be through a womb and birth. He would simply show up as Elijah and declare himself to be Elijah.
Instead of the disciples thinking Elijah was reincarnated they just thought that John the Baptist fulfilled the prophecy instead of Elijah, no?