Well, keep the low end clean but drive the mids and upper frequencies. Some bass chorus and phaser units do a similar thing, and some bass compressors split the frequency range and have separate compressors for lows and highs.

You could also look at different combinations of LPF/HPF frequency adjustment and clean/drive mix circuits to tune things even further, but sometimes you can overdo the adjustment thing to no real benefit. However, it is something you could quickly set up in a DAW to test things, where you've got easy access to filters and overdrive pedal emulations etc. You can then easily set up parallel paths and see what frequency splits and mixing clean/and dirty levels does to the sound without having to solder anything at all.