whats the valves look like?? oil maybe coming from the valve end? Outside chance that the oil ring is stuck, but I dought it. Try with the piston up using carb. cleaner and wire brush either by hand or on a wheel. brush all the carbon off the top of the piston, groove, and valve area (valves closed) Wash out between the piston and cylinder walls Sounds like it could be the oil ring to me if it has reasonable good compression. Not much you can do about that especially if there is slop between the piston and bore.

With the head removed, if you can push the piston back and forth with your thumb, then cylinder bore is worn and there isn't much you can do. Short block? Rebore cylinder and replace piston & rings? Either may not be cost-effective.