Suppose the Haven card were rewritten as follows:
+1 Card, +1 Action. Place one card from your hand on top of your deck. At the start of your next turn, +1 Card.
Most of the time, this rewritten Haven will have the same effect. One obvious difference is when you play Haven, put a card back, and then subsequently play a card that draws from the deck in one way (e.g., Smithy) or another (e.g., Loan). Then you'd draw the card you put back, instead of it remaining "set aside" as it would be with the way Haven is supposed to work.
But suppose that after you play this revised Haven, no further cards are drawn from your deck for the remainder of your turn.
How many different circumstances can you think of where the course of the game might still be changed by this revised Haven in a solitaire game? (I can only think of one, but there may well be others.)
As a bonus question, answer the same question for a non-solitaire game.
Edit: Don't read further if you want to solve this on your own. Solutions in replies may appear in plain text.