All that can presently (May 2013) be gleaned about by internet searching is that Carl D. Bethel is the one-off composer of That Moaning Trombone (1918).
Surfers are welcome to contribute information about this composer.

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Taking up the challenge in January 2017, Rick Benjamin, Founder & Artistic Director of the Paragon Ragtime Orchestra, kindly sent me a newspaper cutting with article given in The Independent Press of Bloomfield, New Jersey on 1st July 1938,
Here Carl D. Bethel is described as a graduate of the Ohio University of Music. He had been a member of the Sousa, Pryor and Conway Bands. He directed his own band at the Steel Pier in Atlantic City which later played in Venice, California.
He had been a director of the Newark Philharmonic Concert Band and had conducted bands for the United States Automobile Corporation, the General Motors Corporation and the International Harvester Company of America. Residing in West Orange, he was made instructor of the Bloomfield Post Office Band.
Thanks a million Rick!