Things that let you stay cool while moving:
Melon hat - slows overheating
Straw hat - slows overheating
being wet - slows overheating
Parasol - slows overheating
Staying under trees where possible -slows overheating
Chilled Amulet - no overheating, must be micromanaged or you freeze
Ice Cube hat - no overheating, makes you wet
Thermal stone that has been placed in either a icebox or near an endothermic fire - no overheating till it runs out
Eating ice -reverse overheating progress temporarily
Using a luxury fan -reverse overheating progress temporarily