I own a terrific Tilley hat. I didn't take it on my first Camino, and I won't take it on my second this fall.
Tilley hats are the greatest, but they're heavy and I don't need the greatest hat in the world on the Camino. When we walked the CF in 2021, I left my hat three different places, and through an incredible number of "the camino will provide" miracles, it found its way back to me each time. I am going to take an old, cheap ($10?) wide-brimmed hat, at least 10 years old, that came from Wal-Mart via my now 96-year-old father, which has seen so much sunlight on summer hikes in the US Southwest that the top of the hat is about 10 shades lighter than the underside. Assuming I don't lose it along the way, I will leave it in Madrid when I fly home with my thanks for many years of underpriced service.
My theory is it needs to fit, needs to be well-ventilated (we're going to be walking in significant heat), and provide good shade. That's it. If it's raining, I'll rely on my rain jacket hood.
I'm happy to spend on great gear for the Camino, but a hat isn't one of those categories.