Parenthetically, it should be pointed out that the 1861 Remington was (and still is) commonly called the "1858" Remington because of the patent date on the barrel. It was, however, adopted as the 1861 Remington New Model, per the above post by TommyGunn.
Just to avoid possible confusion on these "model" dates.
I had a replica of one of these, and it is one beautiful work of art, in my opinion, with the bright blue steel contrasting with the brass frame.
With respect to how extra cylinders might have been carried, it was common in the "long rifle" muzzle loading days to have what was called a "possibles bag." This was a slung pouch which carried extra balls, patches, flints, etc. In other words, everything the frontiersman might "possibly" need as far as shooting was concerned, although the powder flask (or horn) was usually carried separately.
I doubt that there was any formalization of the layout or construction of the possibles bag, and I don't know if the term was still used in the mid-19th century.
Terry, 230RN
REF:
American Rifleman, "I have this old gun," December 2013 p. 112