I've noticed that I can have a fun and organic encounter with a girl, but then when it comes time to get her number, it always seems a bit contrived, and the romance of it dies a bit. The girl knows: "Okay, it's official, this guy is gaming and hitting on me, he's not just some awesome guy."

So what are your ways to get her phone number/contact info without disrupting the smoothness of an interaction?

One I've thought of is somehow playing with each other's phone and then calling yourself. Or maybe writing your number down and giving it to her, or giving her a business card. That way there's not that moment of neediness.