Start with the snare drum since it is one of the most prominent drums in the mix. The snare is also one of the easiest to muffle. Place one of the O-rings on the snare head and let the engineer hear what it sounds like in the booth. If there is still too much ringing you can use the second ring as well.
Remove the front head of the bass drum. Place the pillow up against the head and check the sound in the booth. If there is not enough muffling you can add a second pillow or fold the first pillow in half. Once the drum is suitably muffled, you can put the front head back on or leave it off.
Tune the toms to the desired sound. If the toms sound good without muffling in the engineering room, leave them alone. They can be recorded and then tweaked with some E.Q. If there is excessive ring put a little duct tap on the head at the top. Use about one square inch. If that doesn't do the trick try a second piece of tape on the bottom. If this still doesn't help, tape a cotton ball or a piece of gauze to the head.
Finish the rest of the toms in the same manner as the first. Make sure all the toms are muffled the same way.
Listen to the drums as a whole in the sound booth, or have your engineer listen. If the set sounds good together, you are ready to record.