What most jazz pianists after ~1955 did was to play left hand voicings which often omits the root of the chord, since the bass player would play it. If you have a Cmaj7 chord, the most basic way to play it with an imaginary bass would be the third and the seventh (E and B). There´s alot of ways to vary this but the basic is to play the 3rd and 7th in every chord.
If you would play this voicing and add a C in the bass on the first beat, the chord with E and B on the 2nd, then G in the bass on the 3rd beat and E and B on the 4th , you´d have a basic idea of stride piano (Art Tatum, Willie the Lion Smith, Fats Walleretc ). Like in ragtime. . Pianists like Bud Powell and Thelonius Monk often played some crude souding left hand voicings, mostly for rhytmic accents. Over a G dominant seven chord Bud Powell sometimes plays a G and Ab seperated by a minor ninth which is really dissonant but works in the context. Then the walking bassline like you mentioned.
Then you have all the forms of arpeggios and broken chords for left hand accompaniments which are common in classical music, like alberti bass (C,G,E,G in a C major chord) that you hear alot in Mozart.
You kind find charts of arpeggios over basic major and minor chords
here. I hope this helps some.