As a habitual wearer of a wide-brimmed hat (a Tilly, to be precise) who had occasion to go bare-headed (not counting hair) at times recently in both rain and sun, I know I am much happier with headgear under both conditions, even without being cave-adapted¹ or genetically predisposed to being troglodytic.
A good piece of headgear should indeed mitigate various possibly unhappy effects (depending upon the hat it might substitute different thoughts due to the need to keep it held on in more extreme conditions, or to take it off to avoid social or physical issues when inside with low ceilings and high etiquette). Additionally a hat or hood or a helm is a traditionally utilised anonymiser for cultists, assassins, PIs, spies, faceless goon-guards, etc - and I'm sure could be given these additional effects (somehow, the benefits of the covering never being quite countered by the associated disadvantages such as "Hey, that person is trying to look inconspicuous. How conspicuous of them!")
That and and a good cloak or other upperwear (myself, I'm content to have wet shoulders as long as my head is kept sheltered, but my fleece jacket with extra-hydrophobic shoulder-panels still adds greatly to the sense of invulnerability to the elements), with optional boots/gaiters/leggings and/or waders being primary mitigators against sodden terrain and underbrush (post-rainfall, snowfall or maybe flooding, rather than linked to the precipitation itself) there could be a c1400s solution to all this, and maybe even a form of sunglasses (see prehistoric Inuit snow-goggles, or Chinese quartz spectacles from the 12thC that were also used during judicial questioning somewhat like mirrored-shades have been in more recent times) to be deployed against the not-so-pluvial outdoor effects.
¹ I can't say I'm not Server Room-adapted, though.