Could you, after a bit, go over to a water fountain to like get a drink and cup the water and run it over your face/head as though you are trying to cool yourself off . . . and in doing so 'incidentally' wet the top of your head?

It's just an idea that came to me as I read it, I don't know how practical that is. But I know at the gym guys do that all the time, so it's not like it would attract any specific attention.

Is that feasible where you work out?