Although the Gemara doesn’t openly discuss Gilgulim, in retrospect it is obvious that that is what they meant when, in Bava Kama, they spoke of a person becoming an animal after his death. Also, the many Chazals about how this one is also that one and the like are best explained with Gilgul. In fact, this is one area where the rationalists and the Kabbalists are on the same page. The ibn Ezra seems to imply this when he says that Bilaam is not Lavan and that when Chazal say that he is, it is probably a Sod.
Gilgul was not openly discussed by the Ramban. He treated it like a Sod Gadol, and only hinted to it. However, one generation after the Ramban it is an open discussion, possibly because of the Zohar Hakadosh. Rabbeinu Bachya and the Rikanti speak of it openly.
Rabbeinu Saadya Gaon mocked the belief in it. The Rikanti relates that the Ri Sagi Nahor was able to tell previous Gilgulim. The fact that it was such a well guarded secret shows that it wasn’t meant to be understood as a person jumping around from body to body. The Arizal speaks of many types of Gilgulim. Without understanding it properly there will be open Stiros.
The Maharal in Torah Ohr, where Chazal speak of Memuchan being Haman, explains that he was in essence the same person. Reading that might shed light on the whole concept, and how it is meant to be understood.