In all likelihood, your M1860 Spencer Carbine is chambered for the #56 (aka .56-56) cartridge. Note, however, that the .56-.56 refers to the diameter of the cartridge case at the base and at the front end of the original cartridge. The interesting thing is that the rifling is tapered from the breach to the muzzle! Just ahead of the chamber, the groove diameter is around .545" tapering to .535" at the muzzle. At least I have found that to be the case with a number of these carbines. The six-groove rifling indicates it is the original, whereas the later Burnside-made used 3-groove .50 caliber barrels, that were .515" effective groove, similar to what became the barrels of the .50-70 Trapdoor Springfields.
The original cartridges for the .56-56 Spencer used outside lubed heel bullets. I experimented with several bullets, some made by Rapine (Ray retired years ago, and the company went out of business). I also cast hollow-based bullets from a Lyman #533476 mould, using #2 alloy, sized .538" with good results, using either Pyrodex RS or 8.0 gr. Unique. Cases were made from Dixie Gun Works .50-70 brass, shortened, and inside reamed to .525" before annealling about 1/2 the length of the shortened brass.
Be sure to have the carbine checked by a gunsmith who knows what he/she is doing!
Happy Thanksgiving!