Here in Soith Florida is hell on earth for 3 month out of the year. I ride to work mostly very early in the morning and come back at night. Still carry two insulated water bottles with ice and before the 55 min commute, I drink at least one ice water bottle, sometimes two depending the time of day. Drink the cold water every 10-15 min, can go 30 at times if I pre drink a lot. The sweat will keep me kool while moving, it's when I stop at the light, specialy running errands during the middle of the day.

Have not try the ice around the neck in a bandana, but that sounds very good for this heat!