First of all, remember that the CW was fought pretty much with Napoleonic tactics, hence massed fire, etc. Artillery was just becoming a science, and long gun use was still a point-and-shoot drill. The pistol was considered a secondary weapon, and was not a primary arm even w/ the Cavalry. Officers bought their own arms, but even then not too many actually carried pistols.
I have run across only one source that even addresses pistols: this is Gen Phillip St. Geo. Crooke's 1862 "United States Army Cavalry Tactics". Since the sabre was the primary weapon, it goes into great detail on sabre drills and use. Pistols rate a few small sections. The pistol was presented by the left hand, as the right was the sabre hand. A quote from "Target Practice":
"The target is six feet high and two feet wide; a black stripe 3 inches wide is painted at the centre...When practicable a man is placed behind a ball-proof obstacle, within reach by a wand, of the target; with this he points out the position of each hit, immediately after it is made; so that the person firing can see from his place.
When each has finished his shots, his hits are marked, measured, and noted, as well as the misses.
The squad first fires at 20 yards, each man fires to the front, rear, right, and left; one shot each a day.
The firing is next at thirty yards, and is carried regularly by the decimal inrease up to one hundred yards.
An officer is always present at target firing."
Note he did not specifiy whether this was carbine or pistol, but since he ignores the carbine, I assume it's for the pistol.
Some comments from the firing section: "...To cock the pistol rapidly, without the use of the left hand, whilst pressing back the hammer with the right thumb, to assist its action throw forward the muzzle with a quick motion." Sounds like the way old Hollywood shot their 6-shooters.
Summarizing the Load procedure, holding the grip by the left hand, place against the left thigh w/ the butt resting on the saddle.Take a cartridge w/ the right hand, load & ram it. Hold the pistol against the right side of the pommel & cap.
Good luck on your quest for information.