Y'all Come Back Saloon by The Oak Ridge Boys
Another interpretation that just occurred to me is that the bar where the singer sings and Cowboy drinks, IS in fact the Y'all Come Back Saloon. The UNNAMED song that Cowboy requests every night eventually became the "last call" song for the bar, or the "late-night benediction" (which in this interpretation is definitely a description of the song as opposed to the name of the song). Cowboy is still broken-hearted, and to a lesser extent, so is the singer. But perhaps that pain is not stretched out over as many years or across as many states as my other interpretation.