For http://www.hearts-cardgame.com/ I wait until you are into your second game and then I slide down a little, uh, top banner (what are those things called?) which has the text

"Hi there! Looks like you're enjoying the game. That's great! We'd love it if you could help us out by sharing it with your friends:"

And then has the usual facebook/twitter/etc buttons. There is also a discreet "Share with your friends" link that makes the banner pop down, I like that more than having all those ugly buttons visible the whole time. I wait until you're into your second game because I figure by that point you must like the game, otherwise you'd have left, and then it's maybe more likely that you'll help me promote it.

All my online card games (3 of them) also have a "Also try our other games: X and Y" links, which drive a fair bit of traffic between them.