Also try tuning and damping your snare to match the song you're recording. Consider the key of the song as well as the type of music.
Once that's worked out, then start messing with mic placement. For close miking, I like to start with a three-finger distance from the snare. Sometimes it's better to mic the side near the shell rather than above the rim. Other times it's better to point the mic at the drummer's crotch (I know, go ahead and make your jokes... ha ha ha...). Start with the overheads as mentioned above and BLEND the close mike.
Ultimately the "sound" of a snare is a culmination of MANY factors...
Who = the drummer
What = snare type, size, type of heads and the condition those heads are in (metal, wood? 14x6? 10x2?) and tuned to
Where = the room (Room acoustics play an important role, also the relationship of the snare in the room - proximity to other objects ( other drums vs. a nearby couch, walls, ceiling height, carpet, etc. etc.)
How = with what type of force, with what type of stick, where you strike the drum, ghost notes or rim shots, dead center or near the rim, etc. etc.