One thing they used was the iconic cowboy hat. It has two features that help with cooling. The first is the broad brim to provide shading. Cowboys basically grew out of the Mexican adoption of Spanish cattle raising culture. When Americans start flowing into west Texas at the end of the 19th century, they learned from the people that were already there. The brim on a cowboy hat isn't as broad as a sombrero, but it's the same idea, to shade the wearer.

Many hats have a lower crown to trap heat by the wearer's head to keep them warmer. The cowboy hat has a higher crown. This let heat released from the wearer's head escape or at least not be trapped right by the head. 10 Gallon cowboy hats are the most extreme for the cowboy hat style. You also see something similar with the famous cigarillo hats that the caballeros wore (Emiliano Zapata favored this kind of sombrero and now it's often called a Sombrero de Zapata https://www.mexicoescultura.com/galerias/actividades/principal/image79_2.jpg).

Anyway, in 1865 John B. Stetson recognized these needs, saw what the Mexicans were doing and invented his version, The Boss of the Plains hat. B/c it met the needs of cowboys so well they adopted the hat and it became iconic. It had other features that were useful to cowboys, but the brim and the crown were the important things for heat.

The Cowboy Museum has a little video that talks a little about it. https://youtu.be/HNqAzyBfXe0

I would think that they used little tricks to cool down. Drinking lots, staying indoors as far as possible, wearing cowboy hats (maybe?) but most of all in hot climates the infrastructure tends to be adapted to deal with heat.

Structures like verandas really help reduce the amount of heat entering a building.

But I imagine productivity and just general satisfaction with life probably dropped significantly during this period.