I think if you want to carry a single-shot pistol, one that uses the same ball and patch and the same granulation of powder as your rifle makes the most sense. I recall there was a passage in Parkman's book, The Oregon Trail, about the author running buffalo with his rifle, and finding that he only had pistol balls in his bullet pouch. This also implicitly confirms that old-time plainsmen did at least sometimes carry pistols. Ruxton and Wootton also mention the trappers carrying pistols on their persons.
However, as soon as the mountain men and plainsmen could get revolvers, those are what they carried. There are first person accounts of people killing buffalo with their Army revolvers, and Theodore Davis, in 1869, explicitly stated that they preferred the older style revolvers (dragoons) to the newer, lighter Army models because the older style guns held more powder. The point being that you might consider carrying a percussion revolver instead of a single-shot pistol, if it is legal to do so where you are hunting. Holsters that fit the gun and carry well would be easier to find for a revolver. The caliber would probably not be the same as your rifle, but if you load the sixgun in camp and leave the extra balls there, you won't get them mixed up with your rifle balls out in the field. I would think that the five or six rounds in your revolver should be adequate if it is just a back-up gun.
Anyway, in response to the original question, if you carry a single-shot pistol afield as a secondary or back-up gun, I do think having one that uses the same components as your rifle would be a good idea.
Best regards,
Notchy Bob