It protects your hair when you're sleeping, when you have to wear a helmet or rough hat. And if worn right, they look pretty neat. (you can look like a gypsy or a ninja turtle ^_^)
I've tried satin mop caps from walgreens, but they make them with this polyester layer that really holds heat in. (and they sew them inside out)
But today i've found two neat sites for all your scarf needs. They're built for chemo patients, but there's no reason that you can't wear one on a long bike ride:
http://www.4women.com/fabrics/transitional/1#
Happy Shopping