As far as I can tell, nothing but IGG, a played Action card, Secret Chamber's reaction (which requires a played action card), filling your hand during cleanup, or one of a few "at the start of your turn" effects can ever move anything into anyone's hand. IGG is intriguing but it specifically names Copper as the card to be put in your hand.
The one hilarious loophole that has occurred to me is that if you consider the game to end during the Buy phase (though I'm pretty sure it actually ends after Cleanup) then you get your Island mat's contents back at that time... but of course you can't get the Gold to your Island mat either since you'd have to play the Island!
I'm stumped.
pre-edit for ninja'd: A-ha, I wondered if this was some kind of puzzlemaking error