At long last, I've found out who the song is by!

I've recently been buying records of a North Carolina group called The Briarhoppers; when they first formed, they were based out of the radio station WBT in Charlotte, North Carolina, and performed around the area for several years, eventually re-forming with the original members in the late 1970s-80s for a few recordings for posterity.

I bought the record Whitey and Hogan (with the Briarhoppers) Volume 1 (Old Homestead OHS-90089), 1981, and while listening through it, lo and behold, there it was! After all these years, I recognized it instantly.

The song is titled "The Bible My Daddy Left Me." According to the liner notes, it was written by the members of the Briarhoppers in 1947 but was unreleased until the release of the aforementioned record.