The US and CS produced millions of Nitrate paper cartridges during the Civil War, that was the standard method to reload. They were issued in packs of 6 with caps in a package.
I don't think much reloading occurred in the heat of battle. Ramming 6 cartridges and capping would have been a task with all that going on. Maybe behind cover, during a "lull" in combat but not on the move.
Officers and enlisted men issued or authorized a sidearm usually didn't carry a lot of extra rounds anyway. The "standard" was 6 rounds in the revolver cartridge pouch on the belt but I'm sure some guys carried more packs in haversacks or saddle bags. A smart Officer or soldier that was actually going into an engagement would carry several reloads
If you were one of the "non combat " personnel like a Quartermaster you probably didn't even load your revolver .
Dragoons had been reloaded from a flask in the 1840s so a few may still have been loaded that way.
Also like was said, I'm sure there were Irregulars, Guerrillas, Raiders, Home Guard Militia , guys just "helping out " that weren't really military , that used loose balls they cast themselves, powder flasks and a tin of caps to keep a six gun running.
I've read that fat from Pork was used over bullets to keep revolvers running but I suspect that was probably done before the action, not during