The rhymes on this song are mostly from logical auto or identical word rhymes. The first verse sets the pattern:

Long-distance operator
Place this call, it’s not for fun
Long-distance operator
Please, place this call, you know it’s not for fun
I gotta get a message to my baby
You know, she’s not just anyone

“operator” and “fun” just repeat to summon the rhyming sound.  Need a rhyme? Just repeat the word, right?  Easy?  Well, there’s an exception at the end of each verse.  Note the last word in verse one, “anyone” rhymes with the second repeated word rhyme, “fun”/”fun.”  “wait,” “inside,” and “wire,” are the other non-identical terminal word rhymes, the best I think being “wire” with “higher.”

An amusing thought to this repeated rhyming is how often you have to repeat words when speaking to an operator.  Perhaps a little shot at those frustrating calls from Dylan’s rhyming arsenal?  A stretch right, but it never hurts to speculate what Bobby D has up his sleeve when it comes the to purpose of his rhymes.

Richard Manual is the vocal on the version we all know well from The Basement Tapes, but I found a cool cover of it by the James Cotton Blues Band from 1970–dig the harp!: