Tommy McLain is having a moment—a bright moment in the classic swamp pop singer’s long life of golden ups and disastrous downs. North Carolina’s Yep Roc Records has just released McLain’s new album, his late-coming magnum opus I Ran Down Every Dream.

A lifetime ago, in the summer of 1966, McLain’s feverishly spellbinding remake of Don Gibson’s heartbreak ballad, “Sweet Dreams,” peaked at number 15 on the Billboar...