You ask if they had simpler uniforms. One instance I can think of was a headdress called the Havelock briefly used by Union soldiers at the start of the war. During the Sepoy Rebellion, Sir Henry Havelock of the British Army created a covering intended to be draped over a soldier's cap. It was a white linen headdress that covered the neck, as if each man's hat had a white cape.

It was effective in preventing sunburn, but the Havelock fad meant many were created by well-meaning women behind the lines who hadn't the slightest idea of what they were doing. Many were far too heavy and blocked the circulation of air around the head, thus making their users even more prone to sunstroke.

As Mary Livermore recalled in her memoir My Story of the War in 1888:

"In the early inexperience of that time, whenever regiments were in camp awaiting marching orders, it was the custom of many women to pay them visits, laden with indigestible dainties. These they furnished in such profusion, that the 'boys' were rarely without the means of obtaining a 'permit' to the hospital until they broke up camp. While the Havelock fever was at its height, the Nineteenth Illinois, commanded by Colonel [John Basil] Turchin, was mustered in, and was ordered to rendezvous at Camp Douglas.

A detachment of the 'cake and pie brigade,' as the rollicking fellows called them, paid the regiment an early visit, and were received by the men who were not under drill, en Havelock. As the sturdy fellows emerged from their tents, all wearing 'the white nightcaps,' as they had irreverently christened the ugly headdress, their appearance was so ludicrous that a shout went up from officers, soldiers, and lady visitors. They were worn in every imaginable fashion,─as nightcaps, turbans, sunbonnets, bandages, sunshades,─and the fate of the Havelock was sealed. No more time nor money was wasted in their useless manufacture."

You raise a good point, and it is something that was definitely addressed during the ACW. A couple of things come to mind from the books I've read. In 'Mr. Lincoln's Army' historian Bruce Catton spends a good deal of time talking about the drilling that took place early on in the war (especially after McClellan took command). As you mentioned, marching about in wool uniforms would not be especially comfortable, and would definitely tire a person out/warm them up, yet drilling allowed for 2 things. First, it conditioned the soldiers, and helped them build up a tolerance, as well as expectation for, the heat they endured during marches. Over time, the soldiers learned how to pace themselves, and got into what you might call "marching shape." 2nd, the drilling allowed the commanders the opportunity to learn how hard they could push their men, and gave them the hands-on experience to better gauge how far/hard they could push their men. In other words, after time, regimental, battalion, and even division commanders learned how hard they could push their troops (and under what conditions). Of course, this also developed further during campaigns, on marches, but through drilling, the troops and their commanders developed a good foundation.

Two other things. You'd want to keep in mind that the standard practice of the day was to march 45 minutes, and then break for 15 minutes. This allowed the troops to exert themselves a bit, and maybe even get a little hotter than they would like, but then rest for a good chunk of time every hour. Good commanders were those who used their experience during camp drills to accurately gauge their soldiers' level of fatigue, as just mentioned, but they also would have been good at picking smart places to stop (like when passing a fresh source of water). If it was a particularly hot Virginia day, and the troops seemed especially worn out, a good commander would have probably called breaks at 40 min. instead of 45, and made sure to keep his boys hydrated.

Lastly, commanders very rarely put their soldiers into a fight directly after a march. I mean, it did happen (A.P. Hill at Antietam is a great example), but after a good 10, 15, or even 20 mile march, especially if it was during the summer, soldiers were usually pretty used up and combat ineffective. So to your point, the heat and the wool uniforms definitely led to some problems, but like most good soldiers during wartime, the troops learned to live with it, and their commanders (usually) learned how to temper the disadvantage.

[Further reading: Bruce Catton, 'Grant Moves South'; James McPherson, 'Battle Cry of Freedom'; Shelby Foote, 'The Civil War, A Narrative, vol. 2']