Yeah, while it's not spelled out exactly
how she came to interact with the Philosopher's Stone, it's clear that somewhere during the first five-day loop she had at the Tower (which is
not the same as the first loop that we
see in the game--that's important to remember, that the first "Night Before" and everything after it that we see are actually hundreds or thousands of loops after the prologue--it's just that Lillet is coming out of a "made it to the Stone chamber, lost her memory on reset" event as is described in that final scene) she managed to link up with the Philosopher's Stone in some way.
The particular mechanism by which she was able to do that is unknown, though--did she somehow find the secret door in her room and get to the Stone chamber? If so, how, when in Loop V she has to have a golem tear that door open and then fight her way through Calvaros's horde of enemies left to guard the passage (that would have easily defeated the apprentice Lillet originally was), could she have gotten there? Did somebody like Gammel, Advocat, or Lujei help her? Or was there a way by which she made the connection to the Stone without actually entering its physical presence?
...Sounds like theory or fanfic fodder, to me!