There are very few situations where making a move on a woman who is attached (even if it's a first date) can be justified. It's pretty much a bad idea all around with some very specific "if"s. Here's one:

 If the woman is patronizing a complete loser, there may be an opportunity for you to offer her a better option. This option is only available when a) the woman's date is obviously annoying and her interest is elsewhere, b) the aforementioned loser is not present (watch for bathroom breaks), and c) you yourself are on top of your game as far as wardrobe and general hygiene. This should be handled with the utmost tact and subtlety. If you are prone to ending conversations with a foot in your mouth, it's best not to try this move.

Humor is the best weapon against the ice of the unknown. Write a very short, witty line on your business card (a napkin also works, but with not as much impact and much more cliche'). Drop it in her immediate view (not directly in front of her; let her curiosity do the work), and shoot her a quick glance as you leave.