The traces of spots left behind are probably some shallow pits in the steel, created by the patination process. So long as they're not too deep, just doing some more polishing over time will probably reduce them or maybe even eliminate them. That's probably the safest, most conservative approach.
You might try the white compound (assuming it's aluminum oxide) on a firmer substrate for your strop. A firmer substrate will make the compound work more aggressively (faster) that it would on a strop of leather. A firm strop of balsa or hardwood works for that. And other compounds like diamond at 3 microns or finer will work even faster, used in such a way on a hard strop of wood. I've especially liked 3 micron diamond on hardwood for very fast polishing of even very wear-resistant steels like D2 & S30V, so long as the finish on the steel is relatively refined before you start (anything finer than a Fine or EF diamond hone finish, for example).
And as compared to a leather strop, the white compound can also work very fast on a hard-backed strop of denim or linen (or canvas), all of which will take and hold a very dense application of the white compound. That makes it a very aggressive polisher as well, and it will even thin & shape the steel near the edge (to convex) when used on relatively simpler steels.