I can't put my finger on the book right now, but IIRC there are several photos of cowboys up Montana-way (1880's - 1910 or so), wearing a sixgun in a crossdraw holster. First of all, there could be problems with your horse throwing you, a foot stuck in the stirrup, and the sixgun might save your life by shooting the horse! Rabid skunks, coyotes, etc., could also require dispatching on the range. (The mountain states sometimes have cycles of rabies in these animals. Be sure your pets are vaccinated!)
As to packing two guns, not only was that too expensive for the average cowpoke, but excepting during a rangewar, it might be prudent...if you could afford it. There might be exceptions in the decades following the CW, during the transition from cap-n-ball to cartridge, and depending on where an individual was headed. If a man had acquired a second percussion revolver, he might pack both of them. A number of officers or more affluant veterans might have acquired a smaller caliber rimfire gun, such as S&W #1's in .22 short rimfire, or a #2 in .32 short rimfire as a backup gun, or perhaps some other cheap pocket gun. Don't forget, even a small caliber projectile was respected, nearly as much as a .44 or .45. It was not a question of stopping power, but fear of sometimes fatal infection that could result from a gunshot wound of anykind. In the CW more people died from lockjaw (tetanus) than any other cause except maybe cholera or typhus. (It was worse in the Eastern battlefields because the grounds were polluted with tetanus from horses and cattle feces. But you could still get a fatal case of "lead poisoning" from the dirt and debris carried into a wound, even by a small bullet or ball!
Not a pleasant way to die. )
So, was it common for a cowboy to pack two guns? No, and if the outfit he worked for forbade it, none, but never say "Never!" and don't say "Always!" either.