Rush has a song called ‘Anagram (for Mongo)’ in which the primary words in most lines are anagrams of each other. Fun song but apparently written just for the wordplay.
Hair of the dog is a pretty established idiom - based on the notion of ‘like cures like’ - i.e. that a hair from a dog would cure a bite from that same dog - it goes all the way back to the Romans. In more recent times, the phrase has been accepted to mean a drink of something to cure a hangover from drinking a lot of the same thing the previous night.
Maybe not a pun as much as an extended metaphor with a few good puns tossed in … Sniper by Harry Chapin does a lot with comparing a conversation to a sniper shooting at people. The sniper, ignored his entire life, finds a way to make people listen to his questions. When he starts shooting, “his pointed questions [are] sprayed”. When the cops take him down at the end, “with their final fusillade, his answer had come”, fusillade being both “a rapid verbal outburst” and “burst of gunfire”. Pretty cool song.
The blues song* “The Sky Is Crying” actually milks the old Hendrix “kiss this guy” mondegreen (intentionally or not) by repeating the lyric “The sky is crying” throughout, then at the end changing it to “This guy is crying.”
*forgive me if its original form is otherwise, I only actually know it from Stevie Ray.
"The World’s Address ", by They Might Be Giants, which not only has a pun as its title but comments self-referentially on its own paranomasia in the chorus.
Elvis Costello songs often contain a fair amount of wordplay, which is sometimes cool and sometimes annoying. A favorite of mine is “High Fidelity”… a song about relationship faithfulness which both gives new meaning to the term and deftly references the usual meaning (“Can you hear me?”).