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Wearing shit around your neck is uncomfortable in direct sun, but has a few benefits as having a useful utility rag and you can wet it on something that stays wet like cotton and have that cool you.
A lot of ball caps though do not have a long enough brim to shield your eyes well. Pic related is one of the older style. They are wider (covering peripheral) and longer. I know a lot of people pair the ball cap and buff/neck cover.
The OR sun hat does really well. Sun can still be somewhat strong on back of head. Brim is too short.
The goofy hats you can find on outdoorgearlab where the front brim is stiff and the back brim is floppy and rests on your neck are a great design, but hideous.
For most cloth brim hats like boonies, they share the same issue with caps where the brim length is too short. I think you'd want 3.5" which is hard to find for cloth. Some boonies are stiff enough that you can shape them to shield your eyes better. They do sell hat stiffener fabric spray.
Hard to beat a straw hat for summer. I think palm leaf is a superior material. A lot of these hats do not sustain rain well and will look like shit after a season, but can keep pumping, are cheap, and the brim length can be made very long
When things get cooler, you have to go back to your roots of the beaver and go with a felt hat like we have used forever. A stetson famously was left in the ocean for like a decade or something and recovered and was still fine. I really like Akubra, which uses rabbit felt. Made in Australia. You can get the sombrero if you can bash it well and have immense brim length. The riverina does well too if you aren't as good at bashing, since an unbashed sombrero looks goofy. The main con with these is you will have a lot harder time wearing them sitting down. I would rank them as follows
1) Felt
2) Straw type
3) Goofy outdoorgear lab
4) Outdoor research
5) vintage trucker hat/neck cover
6) baseball cap
7) durka durka shemagh
Have not tried a sun hoody but made of polyester