Caveat: my base kit always includes an Arcteryx hooded rain jacket.
Cold weather (<20F)? Knitted wool beanie. I make my own, and if it is raining I raise my hood over it.
Monsoon rains? Raise the hood.
All other weather? Pure beaver cowboy-style hat. These are extremely hard to come by and are usually very expensive, but they retain form and shape extremely well even when soaked, provide great shade, and are very breathable. For anons in the SW, head to Virginia City, NV, and once you're done drowning yourself in Bloody Marys and visiting the tourist trap shops, head to the Virginia City Hat Maker (Les Chapeaux Boboulin). Do not select the cheapest option. While his 80% beavers are perfectly serviceable, for around $600 you can get a 100% beaver hat custom made for your head while you wait, any style you want, and the quality is such that your grandchildren could wear it with a rebuild every 20 years or so. Go for a >3" brim to keep the elements off of you and provide shade. I wear mine doing ranch work and hiking and it is absolutely wonderful in any weather condition that doesn't involve extreme winds (which can blow it off your head) or cold (it keeps your head warm, but doesn't protect the tips of your ears).
Added bonus: the hatmaker, a cool as fuck French dude, offers lifetime maintenance on your hat. Get it dirty? He'll clean it for free. Damage the lining or sweatband? He'll rebuild it for free. Want it reshaped? Yeah, that too. Most hats you find in western stores are either wool felt (garbage) or hare felt (short life span). Even those advertised as beaver hats have so little beaver pelt in them they might as well have waved a carcass at the hat. To get a mass-manufactured high-content beaver hat usually runs $2500+. This dude will literally style one in any fashion you want (he offers input to make sure it will look good based on your face and build), and throws in a custom leather band for free.