Currently when you collect a Jigsaw Piece you are automatically warped out of the Secret Level
There is speculation (link) that this didn’t used to be the case, and you used to have to exit the level through other means, such as a Door or Swap Meet Pete Gong
- in Sailor Jigsaw the level wraps back around and you collect the Jigsaw Piece by the entrance
- in Caveman Jigsaw there is a trick Swap Meet Pete Gong before the Jigsaw Piece, which you presumably had to go back and hit
- in Inca Jigsaw in the room with the Jigsaw Piece, behind one of the walls there are remnants of a door
Apparently if you grab the jigsaw piece while dying you aren’t warped!? (video)