There does seem to be a hint here:
From "Ten Years in the Saddle" (Memoirs of W. W. Averell, edited by E. K. Eckert and N. J. Amato: San Rafael CA, Presidio Press 197:
“When the old dragoon revolver was first made, a dozen of them were sent to his regiment in for trial…each pistol had two cylinders, both of which were kept loaded so that when the one in the pistol had been exhausted it could be removed and replaced by the extra one in a moment. Now said the Colonel [named Harney] (warmly), “No man can be such a blankety-blank fool as not to see the importance of not losing the extra cylinder, for if lost you will have only six shots…” (88-89)
Also, cased sets came with an extra cylinder. Did less fancy ones? That I don't know...and it might be fun to research.
Now...how common was the practice? Who can really say. Cav might have preferred to carry two or more pistols because they had a bit less weight restriction than infantry - although they were still very limited in the extras they could load down their mounts with. Line grunts for infantry often threw away pistols on the march to reduce weight. Heck, they threw away all sorts of gear. And in some regiments it seems that the officers forbade the men from carrying pistols. So...the weight and bulk of a pistol v. the weight and bulk of a cylinder.
This is making me think that the actual use of pistols may be somewhat exaggerated in our depictions. We see the posed photos of soldiers with musket, pack, canteen, haversack, bayonet, sword, dirk, bowie knife, a revolver or two and think that is what they carried. We look at photos taken in the field and very little of that is in evidence. Look at photos of cannoneers at drill, and they are unencumbered by almost all of that. Plus we have letters and diary entries telling of soldiers shedding equipment after the first mile of march and making do with very little.