Set a 12-by-12-inch piece of 1/4-inch plywood over the existing hole in the wood door. Secure the plywood to the door by placing one self-locking C-grip above the hole and one self-locking C-grip below the hole. Ensure the grips hold both the plywood and the door in their jaws.
Mark the center of the hole by placing intersecting horizontal and vertical pencil marks on the surface of the plywood. Make sure the pencil marks align with the existing hole.
Secure a hole saw in the chuck of a drill.
Align the tip of the drill bit with the center of the two pencil marks. Pump the drill trigger to slowly send the drill bit into the 1/4-inch plywood. Pull and hold the drill trigger when the hole saw meets the surface of the plywood. Continue drilling until the saw exits the opposite side of the door. Pull the saw from the door.
Release the C-grips. Remove the plywood from the door.