I really wanted to run with the idea that restarting the game does not restart the game and instead puts you into the one and only game with a new avatar (and a new page to control it). That first hint is also really important, but doesn't make sense until people progress further into the game. I also want restarting to be consistent regardless of the method used to restart.
What I could do is change the first level into a level selection area where there's a flower for every level unlocked. Though I also like that players think they completely restart the game until they catch up to the point where they did restart. A checkpoint for after the stepping-stone level could also work though.
It's hard to think of puzzles that can match that final puzzle in the game, but I guess a cross-level puzzle could at least come close. If the game remains popular after the game jam finishes I'll look into expanding it a bit.