This topic has come up a time or two in the past & IMHO, the use of extra cylinders was possibly the intent in the beginning but after use in the field under stressful conditions & mounted on a frightened horse, I would think the idea was abandoned.
If you think about it, there should be an abundance of extra percussion cylinders floating around out there & if there were, they should have the same cylinder number as the pistol that it was issued to but I have as yet, seen 2 cylinders with the same serial number to a pistol or any extra cylinders in any manner of issue.
I have collected pistols of the Civil War since 1967 & never saw a lot of extra cylinders being offered except from cannibalized pistols.
Living here in Guerrilla country in Missouri, you see photos of Reb's wearing anywhere from 2 to 4 pistols on their person. And you know they carried more in their saddlebags.
The pistol was the preferred weapon of the Guerrillas & after allowing their opponents to fire their long range single shot rifles & they then charged pell mell , laying down a withering fire by using all their pistols. There was no time out to stop & replace a cylinder. The Colt would have been impossible to do on a mounted horse having to keep track of the wedge while replacing the cylinder & holding onto the barrel.
The Remington may have been one the closest to the idea of carrying extra cylinders as their's was one of the easiest to replace the cylinder but that would have been difficult. Even at that, you had to drop the rammer to pull out the cylinder pin with the exception of the very early 58 Remington's that had the sliding cylinder pin. This was a problem as a tight fitting holster could pull on the pin when removing the pistol & the cylinder would fall out. They did retrofit a tiny screw into the channel of the rammer which prevented the pin from being pulled out without dropping the rammer. Alas, a good idea for fast reloading a new cylinder went down the tubes. It is hard to find early 58 Remington's WO this retrofitting.
The closest would have been the Starr which could have been done on a running horse fairly easy but there isn't any evidence there were an abundance of extra cylinders manufactured by them.
And, if the idea of carrying extra cylinders was the norm, I would think that a holder would have been crafted for the gun belt, ala the speed loaders that those of us that carried them on our gun belts when revolvers were the sidearm of the day in law enforcement.
But, you see no evidence of that in photos.
The infantry did not issue sidearms to the grunts & only officers were issued them. That's not to say they weren't carried as soldiers often found battlefield pickups or purchased them. Actually, the Union Cavalry didn't carry sidearms early in the war.
If the truth be known, the most carried revolver in the Civil War was the pocket model .31 cal. & the majority of those was the model 1849 Colt.