So I have been using a couple of hats while hunting the last several years....wool cycling hats! I have one that is plain wool, and the other one has fold down ear warmers, they are warm and the brim is short, and soft so there is no interference with the string even if it touches. I got mine from a woman in Oregon who hand makes them, here name is Randi.
If you search for randi wool cycling caps she will pop up. There are several other places where you can get them. But I liked that these were handmade and I could pick the color (green), she also embroidered an arrowhead on them for me. Just wanted to pass on as these have been great bowhunting hats.