Another option is to EQ out that ring with a very tight dip at that freq. I would bus the snare out to a channel in your DAW and insert an EQ. Then, in order to find our nasty ring freq, take one band, give it a very high Q (so the EQ looks like a very sharp peak) and turn up the gain on that band really high. Now, while the snare is playing, sweep the EQ slowly through the frequency spectrum until you really hear the ring get extremely loud. Once you find that sweet spot, just bring the gain way down. You may have to do this for a few frequencies before you get rid of all the ring. By doing this, you are only affecting those very specific frequencies as opposed to doing a big cut in your mids, which will cut out a lot more than just the ring.
I attached a screenshot of the EQ I used on the Black Beauty to kill the ring. Hope this helps.
