Hydrocortizone is used for redness, swelling, and itch. I dont think it would do anything for pain or cooling relief. Something with menthol in it would help cool.
I have a tube of Walgreens brand Aloe Vera Cooling Gel in my hand right now. It states it releives and soothes pain from sunburn, minor burns, cuts, scrapes, skin irritations, and insect bites.
Cheap too @ 2 for $5.