You've probably seen Headsweats hats in action without knowing their name: they supply a lot of hats used by runners that are emblazoned with some sponsor's logo. Headsweats is often in the background, content to let someone else snag the limelight. They'll just deposit the order check instead.
You can get a Headsweats hat without signing up for a big event or even being a runner, however. They sell through their own site, at athletic stores, and at Amazon. So you can get the benefits of a reflective hat that lets the steam out and weighs next to nothing.
I've been trying out two of their caps as a regular traveler who doesn't run. (And I wouldn't wear one of these to bike because I don't trust the drivers where I live---helmet on at all times.) The first, the Airlite, is a great all-around travel hat. It only weighs two ounces and is easy to pack. The "Eventure" woven fabric makes it far more breathable than most baseball-style caps, which is great because, well, your head sweats when it's hot. The Coolmax fabric stays cool and there's a sweatband where it touches your head.
The bonus is that it's reflective to keep you safe. Even if you're not a runner, your travels will often take you down dark streets and paths where it can't hurt to be seen in the headlights. And if you are somewhere you feel safe biking without a helmet, this will give you extra visibility. And hey, you can just toss this in the washing machine if it needs a clean-up.
The Airlite comes in white with a variety of accents and lists for $20-$23. Buy direct or see if it's in yet at Amazon. You'll also sometimes see an "Ultra Reflective Race Hat," which is even brighter at night.
The Protech hat goes beyond protecting your head and has a built-in flap to keep the sun off your neck. I bought my first hat that did this at least 15 years ago in Bali, so the idea isn't new, but today's wonder fabrics allow a hat like this to be lightweight, wicking, and quick-drying. All great assets for a traveler on the move.
Made of Coolmax fabric, it provides UV protection while wicking away moisture and it's not much heavier than the Airlite. This one also has a Coolmax terrycloth sweatband and you can toss it in the washer or do a sink wash at your guesthouse and it will dry quickly.
I've used this hat a few times while kayaking (like in the photo here) and it has kept me from turning into a redneck. It's also good for the golf course or any hiking trip where you won't be in the shade all the time. One unique feature of this is that the underside of the brim is black, so it absorbs the light from the water instead of reflecting it back in your face.
With it being so lightweight though, it's probably not great for windy conditions, like a powerboat moving at full throttle. Rely on sunscreen for that.
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