There are a few things that work for me that I learned at TIJT in
Do you have end-brushes? They can be trimmed to smaller diameters to
reach hard-to-get spots.
You could also invest in a wood-centered bristle wheel for your
polishing machine. This thing is dandy for polishing up detail
without losing detail. Still, be careful.
Another trick is old-school thrumming. Basically, take some string
that fits through the holes, tack one end to your bench, rub some
polishing compound on the string, and thread it through a tight spot
on the ring. Rub the ring back and forth (don't accidentally make
notches) and the string will smooth the inside areas. Repeat
throughout the ring for a fine polish.
If these were already made and merely tarnished, you could try
pickle, then rubbing the rings with Alberto VO5. This works really
well for things like sterling heishi necklaces.
These last two methods are obviously not good for rush jobs. They
are perfect for a slow rainy day and your favorite music or movie.
If this is a Christmas rush job, as a last resort you might try