As said to end conversations (especially on the phone): mmm-bye.
When and how did this form/usage start?
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MMM is in anticipation of "B".
I find descriptions of the use of "mmm-bye" going back several years and all across the country but most often in the South or warm weather states. The best explanation of it's use and meaning is to say good-bye and close a conversation. It is often used in the general collection of closing phrases such as, "OK, well, we'll talk later, take care, mmm-bye." As stated there is as yet no etymological history of this term.
The use of this extends as far North as North Dakota and has little to do with region. However, in the warm weather (generally Southern) states speech often becomes slower and more relaxed, along with much activity. [This explanation was given to me by several good Southerners.] Clearly the purpose of the mmm is to continue, in anticipation, the use of the B in Bye.
The /b/ being a bilabial stop or plosive consonant requires the lips to be closed to begin with. Rather than anticipate by saying "B-B-B-Bye" the laid back, relaxed method is to start the voice before the lips get into action. These leaves us with the calm "mmm" for as long as comfortable to be finished off with the plosive B, then Y and E, mmmmmmBye. A comfortable and relaxed conclusion.
"Hmm" and "mmm" are, in this case, interchangeable; they both serve as fillers.
There is no proven theory of the etymology of "hmm" or "mmm," but there are some speculative ones.
Regarding the symbolism of "hmm," Anatoly Liberman said:
The first h-sound is simply a substitute for breath, and the second m-sound, since the mouth is closed, is symbolic of the fact that we're not quite sure what to say [Live Science]
It can be reasonably inferred that the same can be applied to "mmm," except for the part about the "h-sound."
In the same Live Science article, Nicholas Christenfeld speculates that "hmm" is popular because of its neutrality. Again, the same could be inferred about "mmm."
The Lexico dictionary defines "hmm" as an interjection used to express reflection or hesitation.
"Mmm bye" was most likely said because "bye" is too abrupt, so "mmm" can act as a filler—for just the right amount of time—which makes "bye" softer/more gentle. It is easy to say because the "m" sound can directly transition into the "b" sound in "bye," (because your lips are closed) which is what makes "mmm bye" smoother than "bye."
Also, saying "mmm" before "bye" gives you a moment to think about whether you want to add anything else to the conversation before hanging up. This is more aligned with its definition than the above use, but I believe that both are just as valid.