I think "Publish or Perish" also reuses the tune from "Ransom for a Dead Man", much in the same way as "A Stitch in Crime" does. There is a scene where Riley Greenleaf (Jack Cassidy), seemingly very drunk, chrashes a party to meet the author and soon-to-be murder victim Mallory (Mickey Spillane) as well as his agent (Mariette Hartley) and his new publisher. In the background, the tune in question is being played on the piano.

"Ransom for a Dead Man", "A Stitch in Crime" and "Publish or Perish", among several other early episodes, all had scores by Billy Goldenberg. Of course, if it is possible to rearrange an old tune for scenes like this, then that is what the composer is going to do (rather than create completely new music every time background music etc. is needed). So it is only natural that music was recycled in Columbo as well as houses.