There was one grey wool cloche I found at a vintage/thrift store about 30 (!) years ago that fit me perfectly, looked quite lovely and that I wore until it was so battered it looked like it had been on the losing end of an argument with a Jack Russell terrier. Since then, I've never found a hat that quite emulated its fit and flattery.
There are two extremes where hats are a must pour moi: when it is very cold, and when it is very hot. Just about the only time I'm in the "cold" extreme is when we go skiing, so I'll spare you the ski caps.
Due to a family history of skin cancer, une femme is unflagging in sunscreen application (even on those days when I'm only going in to the office), but for being outside in the midday sun, a broad-brimmed hat is also a must. There is one I use for working in the yard, a simple white cotton affair with a very broad brim adorned with handpainted flowers that I picked up when visiting the Sherman Library & Gardens in Corona Del Mar. The brim is wire-lined and the hat folds up into a small CD-sized pouch. Great for travel to those tropical locales where much time will be spent on the beach or poolside.
But a stiff wide brim isn't always practical. So I've been searching for some other nice choices for shade hats that are packable/crushable, and here's are a few that look promising:
(of course the hat won't help if you walk around staring at the sun!!)
Shady Lady Hats, "Diva" $55
Travelsmith Cool Cotton Hat, $29
The rest I've come across so far seem to be of the "adventure" hat, cowboy hat or golf hat variety, but the Shady Lady website seems to have a lot of stylish choices. There's no reason one can't still look trés chic while protecting the skin.