I bought a squishable straw hat at WDW years ago - very sophisticated. A Mickey head etched in the straw. It's got a big brim that you can wear folded up or down and it packs flat. It looks cool and it is.
Also a "Tilley Hat" is made specifically to deflect the sun's rays. They're pricey, but well worth the money - and if it gets lost it'll be replaced for free.
Also, ask the pharmacist about a sunscreen that will give you protection, but not irritate your skin. My dd is a pale blonde and with her working outside at Epcot we asked the pharmacist what they could recommend. They gave her a "behind the counter" 60 spf that's worked really well for her. Try it on a patch of your skin before applying it.