In general, your left hand can do three basic things during a solo:
- "comp" (play chords)
- lay out (not play)
When you play chords, you don't want to just play them on the downbeat every time. Finding rhythmic figures for your chords is important. Many people will listen to their favorite recordings and transcribe the left-hand comping rhythms, and then practice playing left-hand chords with those rhythms. This is extremely valuable, but as a shortcut, there are two extremely common rhythmic figures you can use to start off:
You can move these rhythms around so that they start on different beats. For example, you might take the second rhythm and delay the entire figure an eighth note, so that it starts on the "and" of 1. To see a more comprehensive list of comping rhythms you could try, check out the resource at the bottom.
Instead of using a standard rhythmic figure, another option for left-hand comping is to "play in the cracks." That means you play chords with your left hand whenever your right hand isn't playing. Check out some Keith Jarrett for some great examples of this.
The last option for left-hand comping is to play block chords. Whenever your right hand plays a note, your left hand simultaneously plays a chord. You can hear all of these techniques from Bill Evans, but he's especially adept with this. You often hear this sort of technique from Bill near the end of a solo/in one of the final choruses. You can also hear some great examples of this by listening to Red Garland. Here's an example from Bill Evans on Autumn Leaves, starting at ~3:55:
(As a side note, you generally want to play rootless chords in your left hand if there's a bass player. Another good rule of thumb is to stay away from the lower register where the bass player will be playing. That's not a hard and fast rule, but playing too much in the lower range can clash with what the bass player is doing.)
In addition to playing chords, you can improvise with both hands. This could generally take two forms. You could play the same line with both hands (either an octave apart or two octaves apart). For example, check out Phineas Newborn's solo on Oleo:
or Brad Mehldau's solo on Anthropology:
If you really want to go for something aggressive, you can try soloing in both hands at the same time. You can hear Lennie Tristano doing this, or for a more modern example, check out Robert Glasper (starting at 2:17):
This one requires no explanation--simply don't play with your left hand at all! This is most commonly heard at the very beginning of a solo, because when you add your left hand into the mix, it heightens the energy of the song and tends to build the solo. So playing with just your right hand is a nice low-energy way to start a solo.
More possible comping rhythms: