Your great uncle may have been born in Mississippi, but moved to Louisiana before he got his first job. You didn't always get an SSN when you were born, as you do now; you used to get it when you got your first job.
Your first step, if you haven't done so already, would be to get a copy of your grandfather's SSN application, for $27. You might also try for an obituary, too.
My grandfather was a Wilken. He spent hallf his life telling people "Spelled "EN", no "s" on the end." They wrote "Wilkins" anyway. You would be wise, as you keep hunting, to look for "Giles" just in case; I suspect you and all of your family have to say "Jiles with a 'J'" a lot, and you still get written down as "Giles".