Eddy Thomas is NOT a pseudonym of Smith Ballew, as some discographies state it.

Eddy Thomas was a jazz vocalist who worked exclusively for Brunswick during the 1920's and 1930's. He made hundreds of recordings for the company with many bands including Meyer Davis.
Eddy Thomas was also a founding member of the Yacht Club Boys, also recording at Brunswick. He was replaced by another singer in this band in 1930. He probably retired from the musical scene in the early 1930's.