I have an old Navy .36 cap/ball revolver and firing two rounds does not heat up the barrel anywhere enough that would cause the skin to scorch if it was tucked in a waistband.
As was pointed out by a fellow Texan, odds are high that the character would also be wearing appropriate undergarments that would preclude direct contact with the skin.
Furthermore, those old .36 caliber guns were blued as stainless models were still just a few years (decades) away, and blued barrels and cylinders dissipate the heat generated far better than do stainless.
For what it's worth, I first read this question, then earlier today, went to my favorite range and loaded up the cylinder, fired off two quick shots, paused for a few seconds as though I was "taking it all in what I'd just done," then (VERY CAREFULLY) shoved the old revolver in my waistband.
It was warmer than the ambient temperature, but not even close to being uncomfortable.