It is perfectly acceptable to use a minimalistic approach when recording drums (like this). But when you want more options and/or a more modern sound, close mics are the way to go. Lots of them. On everything. Here are some quick tips for tonal flexibility.
Miking the top and bottom of the snare (check for phase issues and perhaps invert the polarity of the bottom mic) gives you body and punch from the top mic AND snare sizzle from underneath. Playing with the blend between these two mics (and EQ) gives the snare a lot of possibilities.
Try using a dynamic mic close to the beater for the attack and a large diaphragm condenser a foot back to capture the deep low end. Blend and EQ to taste.
Using a stereo pair of room mics can provide even more flavor. It is common to compress the devil out of these mics and blend the slammed signal together with the rest of the kit.
Some peak resonances are good and some are bad. Try to find these on the bass and snare with a narrow EQ boost. Sweep the spectrum from low to hi. If you hear a frequency ringing out with a bell-like tone, try cutting that frequency with a very narrow band of EQ and see if you like the sound. This can usually be done one or two places, but if you get too crazy with this technique, you’ll start killing your tone rather than improving it.
5. Buss Compression
Try running the whole kit to the same buss in your DAW, and put an appropriate compressor on that buss to treat everything. I like the Slate Virtual Buss Compressors for this sort of thing.
The main thing is to experiment. Just do it, as the sneaker people say. And compare your sounds with a good target reference sound from your favorite drummer.