"Solitary race" sounds pretty literary. I've never seen it in any other song, folk or otherwise. The author may have picked it up from reading or remembered it from a sermon. (Biblical commentaries of the day described the Essenes as "a solitary race," for example.)

Clay Co., Missouri, was named for Henry Clay. Coincidentally or not, actually used the phrase (in a quite different sense)in a speech in 1829. So maybe the author of "Jesse James" was specially interested in Clay's career.

In the 19th C., "race" often included ideas like "family" from the viewpoint of presumed genetic inheritance. "He came from a solitary race" apparently means that his family had always been more or less loners.

Kind of anticlimactic, if you ask me.