I did an 800 mile pilgrimmage hike in Japan. They wear an all white traditional cotton robe, rice straw hat like that one, straw sandals, wooden staff, etc. I had a white synthetic robe made up, wore trail runners, used trekking poles, etc. For the hat I wore the Chillba and it spent most of the time strapped to my pack. I found it to be hot and the wind was constantly ripping it off of my head. There's also a head harness inside and it was a little snug for my giant head. I thought it would be cool, but it just didn't work for me. Plus it looks goofy so I never wore it anywhere else either.
I prefer a ball cap or boonie hat. North Face and Columbia both make synthetic boonie hats that have mesh around the crown to help with cooling. Not great in the rain, but great in the heat. If you don't mind goofy then you can get a Sunday Afternoons hat. It has a broad brim up front but it's long in the back so that it can be worn with a backpack.