My experience has been that around 106 (shade temp) things get murderous. Stop and rest in the shade.

Desert travelers tend to become nocturnal, especially during the summer. HIke during the night or at least start early in the morning (4 AM). Slow down or stop when it gets hot.

Years ago I worked on a dig near the suspension bridge in Grand Canyon. We started work at 4:30 Am, knocked off at noon, and returned after dinner when the site was in shade. When I hiked to the site, I started down the Kaibab trail at 2 AM.

And yes, a full brimmed, floppy hat with a dark brim is optimum. Cotton works because you can soak it in water and the evaporation will keep you cool (for a bit!)