To re-drill rings I made a simple wood jig for my drill press. A block of 2X4 with a 3/8" deep hole to match the O.D. of the cork ring and a 1/4" dia. thru hole. My rings were 1/2" thick so the 3/8" counter bore left just enough cork sticking out to grab the rings.

I chuck a 1/4" bit into the drill press and use it as a guide to center the jig and clamp it to the press table. I then swap the drill bit for a 1/4" dia. brass tube with a slotted end. The rings are held on center in the counter bore and the brass tube punches out the new center holes.

I found a brass tube worked better in the soft cork, a regular drill bit tends to cause splits if I'm too aggressive.

I got the brass tube at a hobby shop, I used a triangular file to cut a series of notches (like saw teeth) at one end. I use the same tube to cut foam plugs for bass poppers. If you have an old telescoping radio antenna lying around you can disassemble it for several dia. brass tubes...

You could use a hand drill if you used the 1/4" thru hole as a pilot and drilled from the back side.