If there is lots of humidity already in the air, adding misters, watering down the run can add to the misery of the chickens. I am in central Texas and it is hot and humid. Misting the run would add more humidity and if they cannot escape that they will be miserable.
Mine free range and find "cool" (always relative) spots under decks and bushes, etc. I focus on keeping the water cool/cold, I add cold water to their feed, which the love and on days when the weather is over 98 with a high humidity I give them watermelon for extra hydration.
On really hot days I will water the back lawn after the shade hits it so they can enjoy the cool grass while they eat their watermelon.
I also make sure I move their water/food so that they are always in the shade and I change it when it gets warm. No one, especially chickens, like a hot gulp of water on a hot day. Ideal drinking water for chickens on hot days is 50-54 degrees f.