Good ideas, Alex. I find I need a hat with a wide brim all round. One thing I've done is take one of those oblong icy packs, the kind that stay rather 'crunchy,' and put it on my head under a hat with a high crown, with a small cloth in between. For me it seems that keeping my head cool helps with my whole body (and natch with my awful head perspiration). I don't know whether this has any science behind it, but as the saying goes, it works for me.
My physiatrist told me to activate "internal air conditioning", to consume icy foods and beverages that will cause brain freeze.
For example, iced coffee is cold but doesn't cool but if you get a slushy like beverage instead, you will cool down from the inside out.
I'd like to add:
When going from AC to heat be cognizant of transition time.
Transition slowly when possible, for example if working outdoors in the heat, sit in the shade to cool down first, and then transition indoors.
Apply the reverse when coming out of the AC. Go to a shady spot, allow your body to adjust a bit, and then work your way toward those full heated areas.
This tactic has truly transformed my tolerance, and hope it helps you too!