I used this last night and it worked a treat as a chin restraint. It's called a BUFF. I bought one years ago (2007) as a motorcycle helmet liner. I would wet it then put it on and the helmet vents letting in air would provide a good deal of cooling.
It can be used as a light winter hat or a neck scarf. They are highly breathable, light weight and long lived and NO VELCRO.
https://www.buff.com/us/neckwear.html