Here in the U.S. it is caps, and they are generally referred to as gimme caps. They're given out by various businesses and organizations. I have quite a few myself.

I often take a bandana for gardening, and once it gets warm, I drape it over the top of my head so it lies over my neck, then pull my gardening hat over the top of it so I have shade on my face and down my neck.

Probably more than you wanted to know about American impromptu headgear. It isn't a fashion statement, though if you wear the bill of your hat backwards in the wrong neighborhood and your hat is the wrong color, you can get shot.

SRS