I am a mid-19th century historian focused on the period between 1858 and 1865 in Colorado, with a particular focus on the military activities in the southwest in the winter of 1862, so I can speak to the clothing conventions of the working class during the 50s and 60s.

I'll start by saying that yes, 1850s-60s clothing is hot in warm weather, and the people living in the time coped in a few different ways.

When it comes to films in the Western genre, characters portrayed are often working-class Americans. Ranch hands, Bounty Hunters, Gamblers and the like wore ready-made machine-sewn off-the-shelf clothing for the most part as it was affordable and, for the most part, sturdy enough to handle hard labor. Jean cloth was common, but so was cassimere, flannel, broadcloth, linen, and osnaburg. Hats were made of wool felt, straw, oilcoth, fabrics, and rubbercloth. Boots and shoes were made of leather and rubber.

It's important to remember that because 19th century folks didn't have climate-controlled environments, their bodies were differently adjusted to the heat. That being said, they still stripped layers for work, or made clothing choices that were better for warm weather, like choosing flannel overshirts instead of coats, or wearing summerweight coats and jackets made of linen rather than wool broadcloth. There were seasonal clothing choices available, and machine-sewn ready-made off-the-shelf clothing (as opposed to tailored), was the primary clothing supply source of the working class in the 1850s and 60s.

The Gold Rush of 1859 brought a huge influx of working-class Americans to the Pike's Peak region. The mining in Colorado was mostly placer mining, with pans and sluice boxes on stretches of creek bed, with hydraulic systems to blast the banks away. The mining season was the warm season, and it gets hot working in the sun. Miners stripped layers to make themselves more comfortable.

It was common to see men working the mines in only an undershirt, or with a vest over a linen or cotton shirt on hot days. Broad-brimmed hats were a staple on the Frontier. Many day laborers forewent jackets on favor of overshirts, which were mostly unlined and were made of a thinner flannel rather than a broadcloth.

Long sleeves were often rolled up, but long sleeves and a wide brim hat would protect from the sun.

On the Frontier, boots were as popular as shoes. In the mines especially, waterproofed leather or rubber boots were a necessity. The Steamboat Arabia collection contains huge shipments of both boots and shoes, and that was bound for the mining regions of Montana in 1856.

There are misconceptions surrounding what it meant to be "dressed" in Victorian America, and the fact that so many images show men wearing only undershirts or shirtsleeves indicate that this etiquette was looser than it is commonly thought of.

This album contains a variety of images of working class Americans, most of the miners, and shows most of the summerweight options available to the men of the period.

TL;DR, the clothing etiquette of the 19th century working class was not nearly as hard-and-fast as it is often portrayed. Men went without jackets in public, and layers were stripped if folks got too hot. The layering of the 19th century worked well to keep folks warm at night, but during the day and while hard at work, layers were stripped for safety and comfort. There were certainly different clothing conventions than those we have today, but folks' everyday wear required some utility.

You might be interested in the links to similar questions that were answered, posted by /u/searocksandtrees over here.

As a followup to this question, would people have been significantly more mindful of their clothing as a means of protection and less of general comfort?

What I mean is that in today's world and outside of perhaps formal events, we typically dress to be comfortable. That said, we have sunscreens for sun protection, various lotions for sunburns, Neosporin-like creams for mild cuts, and on and on. With these not available during that time period, is there a notion that people would have focused on "safety" (for lack of a better word) over comfort?