And the high heat would also guarantee the water would all evaporate quickly so it wouldn't hang around for too long

Not necessarily. It depends on the weather conditions.

If it's a humid day then the evaporation rate drops quickly,. Adding water on a 90 deg day with 85 deg dew point won't get you much.

All machine efficiency depends on your ∆T. So it will always work better on hot days with cooler water. If it's extremely humid then the evaporation rate will be lower.

Evaporation absorbs the most heat energy. 1cal raises 1g of liquid water 1deg C.
The latent heat required to evaporate 1 g of 100 deg C water to 1g of 100 deg C steam is 540 Cal/g. Same situation with ice to water is 80 Cal/g.

Do you have insulation in your attic? Check the R value for efficiency.

If not, then that helps quite a bit. Also, attic ventilation. Get a attic fan that you can use to increase ventilation. Particularly at night when the outside is much cooler.

Moving air transfers heat away if. Static air insulates (tiny pockets is air in Styrofoam). Sometimes the attic fan works against you. It depends.