In "Double Exposure", Columbo says that he has been "working late on the Hayward case." This refers to Nelson Hayward, the politician who murders his campaign manager in "Candidate For Crime". ("Double Exposure" was originally broadcast as the next Columbo episode after "Candidate For Crime" -- apparently Columbo was working on both cases at the same time.)
"Candidate for Crime" is mentioned again in "Publish Or Perish". Columbo tells killer Riley Greenleaf (Jack Cassidy) that he wants to write a book about his experiences as a policeman. As an example of his potential book material, Columbo describes the plot of "Candidate For Crime". (Then Greenleaf tells him, "Lieutenant, very frankly, I don't give a **** about your Senator or your story.")