Well, as usual, first and foremost it's the actual snare and the guy who plays it and the room where it's being played. Other than that, not being there, I can only advise to try different mics (I'm not a big SM57 fan on snare). I had great luck with the Octava 219 (cheap!) mic and a Neve-ish mi-pre (SCA N72) or a fast mic-pre like the John Hardy M1 (SCA 99 or C84). If you can find a Trident A-range, that would be very cool. The BTW snare was recorded with a MCI 636 console and the bult in mic-pre and EQ. Can't remember the mic.
The reverb used on that mix on the snare was a Quantec QRS and the old (blue) Lexicon 224. We also used an empty oil tank which was under the studio. I can't remember the model of the snare Stephan had at the time, but I would assume that it was a Ludwig.
I would also suggest to set up a second mic (Beyer M201 or SM7) and run it through a tube mic pre on stunn (I use the old GrooveTubes MP-1 for that or the Summit tube mic pre). Make sure it's in phase with the other mic (IBP maybe). Give it a good kick in the gain department, so it overdrives the mic-pre. You might have to tame it at the output to get your recording level in check. Shave some of the high end off that track and mix it in with the original snare maybe at 75%.
Hope that gets you going.