I think before you bet on the turn, you should think about how likely it is that he's got you beat. I'll leave it to others who have a better idea what they're talking about to get into exactly what his range is here, but if you're thinking he might have had a flush draw, you have to figure a big raise is coming if you bet here. If you think he might be weak, go ahead and bet bigger. The problem with the $0.60 bet is that you're still pot committed if he raises you, and you don't have the fold equity you'd get on a bigger bet.

I might check the turn and see what he does. Of course, he might shove all in, in which case you're still in pretty much the same spot, but you have a little more information without having invested the extra money.