Every rifle barrel has it's own little quirks. First you didn't say what kind of rifling, twist and configuration of rifling makes a big difference. Older style rifling with deeper grooves usually does better with softer lead, while shallow grooves tend to do better with harder alloy bullets or even copper jackets. When shooting black powder, I would not shoot copper jacketed anything. The tolerances for copper jackets do not accomodate blak powder fouling vry well. Lead does a better job with fouling. As another said, lube should be appropriate for black powder. Alox is terrible for black powder.
You need a bullet that fills the rifling without too much distortion, so slugging is very important for any bullet. In addition, free bore ahead of the chamber can destroy accuracy. If your chamber allows it, try seating the bullets out far enough so that the bullet is up to or even partially into the rifling when chambered. (Next to impossible with very hard or jacketed bullets.)
For instance micro-groove rifling almost universally requires a hard alloy bullet. An old original may have the soft metal rifling swaged smooth by shooting copper jacket bullets. (As was done by shooting copper jacket bullets in old Trapdoors.)
Your question entails much more than, what to shoot in a 38-55. A modern steel microgroove barrel with close tolerances will probably do terrible with black powder and soft lead bullets. Same is true for an old deep rifled soft steel barrel using hard copper jacketed bullets. My old trapdoors prefer pure lead bullets. My rebarreled hepburn has old style black powder rifling and also doess better with soft lead bullets. My original Martini Henry rifles have choked rifle bores and shooting hard alloy bullets can be dangerous. Soft lead bullets have no problem being swaged down a few thousandths as they pass through the chokes. Hard bullets may cause ruptured barrels when they hit one of the chokes. It will treat it as a bore obstruction.
Lastly, some rifling is great so long as the bore is swabbed between shots, some will perform fair to better with out swabbing between shots. The necessity of swabbing the bore can be affected is you find a good lube that keeps the black powder fouling soft so it is scraped out by the next shot. Just as in muzzleloading, consistency from shot to shot is the key, so if fouling is the same from shot to shot, accuracy will be better than when it builds up over a string of shots.