I've been running on the treadmill at the gym a few times a week and, when I do, I get my cotton shirt about 70% saturated (gross, I know). When I leave the gym, I toss on a fleece over the cotton t-shirt. 5 minutes later, I'm home and too hot in the 62* house to wear the fleece. Once removed, the cotton t-shirt begins to chill me pretty quickly, even with body temp up from the run, 62* ambient and no wind. If I take it off and stand around making the morning brew in just my shorts, I'm much warmer than with the wet shirt on.

All this to say, if you're wearing cotton and you get wet, I think you're better off removing it entirely. If I recall correctly, the aboriginals of Australia weathered generations of winters unclothed next to fires...