From my bits of research, I think the Marcot photo is pretty representative of how the Carbine was slung; I'd love to hear some more comments about if these came with the carbines when issued. For a regular, non-horesback shooter like me, this is what I'd do.
The question is, is anyone making repros of these? If you can find someone, then great. If not, I'll make one up for you. If you send a Springfield sling, I'll modify it into a Spencer carbine sling like the one in the Marcot photo. If this is of interest, then feel free to contact me off the list. They don't look that hard to modify.
On the other hand, you can always purchase the Cavalry sling, complete with swivel to attach to the carbine ring, and sling it like a Sharps. It's a little bulky, and not as convenient, but very cool in its own way.
The issue Carbine Slings can be had from a number of sources, including S&S Firearms.