"and you really only want hints of it." What's wrong with full-on stride? Or using close-voiced four-to-the-bar chords? Or "mini-stride"? (Like stride but not moving the hand, only playing a chord broken up in the middle) Or mashing octaves/6ths in the left hand?
Even pre-bop, like singing Great American Songbook stuff while playing the piano, requires this. Like, for the head while you're singing you can get by on bass and chords while you sing the melody, but when it's time for your piano solo, you're basically trying to fit three responsibilities in two hands. Stride is only one way to do that, and you really only want hints of it.