I learned a lot from jamming 1 on 1 with a drummer I know and since I was the only instrument I learned a technique that had to fill all the space to keep it interesting. So I tend to play a lot of double-stopped inverted 5th chords on the top two strings above a more typical rhythmic bass line on the bottom two. I'll do this low down on the neck, but I play with a pick and usually have my tone fairly middy-to-trebly, and I don't usually play the bass notes and the chords on the same beats but rather match my bass notes with the kick-drum (or complimentary to it) and then put the chords on accent hits on cymbals or the snare. I'll still do this in a band setting; I play with a guitarist and occasionally a keyboardist which means I have to pay attention to what they're doing and not compete or clash with their chords, but generally root-fifths are okay and if I can find where they're putting 3rds and 7ths I'll go for it.
What I do a lot more in the band, because I tend to be a little bit of the musical leader, is I'll play big chords by letting notes I play sustain over the top of each other. That way I show the guitarist exactly what key and progression I'm going for, and it can be really gorgeous and huge sounding. I've found that if you attack each chord tone separately it helps avoid a lot of the mud even if you're playing lower down and with a warmer tone. It's very organ-esque. Other times, for loud rock stuff, if the guitarist wants to solo I'll play a rhythm guitar part up on the top 3 strings around 12th fret to be able to get chords that fill up both bass and guitar space well without being too muddy.
The final trick I use a lot is keeping the harmonic overtone series in mind. Bass guitar plays a lot more note per note than guitars. Each low note we play (not so much past ~7-12 fret on the higher strings) will octave itself, plus sound the fifth in that octave above fairly loudly, depending on how you attack the notes. There's more, but after that fifth they fade away enough I don't really give consideration to it. Those first two overtones really affect your sound though; it's like every note a bassist plays, if they're tone is middy-trebly enough to show it, is an inverted power-chord. If you know how all the fifth intervals in your scale or key relate to each other you can play full, massive sounding, 4-note chords by picking just two notes.
