For contemporary tunes with a swing feel, I think a more pianistic approach works better than the Freddie Green “four on the floor” style. I like to break down my comping teaching to rhythmic ideas, voicing, then chord substitutions and more. Here are some ideas to get started.
First, download the pdf file for this topic. Then, watch the video here. If you have the iReal Pro app, use that to practice on tunes you know or are learning (use the mixer to turn the piano down.)