In the card game a player who has their gift stolen DOES immediately go next, just as in the traditional gift exchange game. When a new gift is selected (a new card turned over) THEN the next player in clockwise order takes their turn to select a gift.
Thanks for clarifying that - it does mean we played our first half a dozen games wrong (it still worked okay though). I suppose if you're familiar with the traditional White Elephant gift exchange this may be obvious, but it wasn't entirely obvious to us from the rules. We've already got in another game playing this correctly, and look forward to trying it some more this way!