I have data on Jefferson and some other possibly true information. He died February 1873 and is buried in Phelps Cemetery, near Paris in Henry Co., TN. His gravestone states that he was born 9 August 1808 or 1809. My information says he was probably the son of Jesse Alexander and Margaret Dodd, but I have no proof.
You might check Henry County, TN, wills and estate papers for Jesse, or Jesse's sons, Myers Alexander, who married Frances Walker, and Joseph Alexander, who married Flora Shaw. There may be references to Jefferson.
Don't get confused between Jefferson Alexander and Jefferson D. Alexander, his neighbor. I'll post more later, if you want it.
Paris newspaper, Weekly Intelligencer, February 27, 1873: JEFF ALEXANDER died at his mill near Boydsville, Feb. 25, 1873 "from injuries received by being caught in the machinery of a saw mill."
WHIG- TRIBUNE, Jackson, March 8, 1873: "Mr. Jefferson Alexander, an old and much esteemed citizen of Henry County, recently while cleaning his saw mill of dust with an ordinary wooden rake, let the handle get too much mixed with the saw, and it was cut into three pieces, one of which was thrown back, striking him in the abdomen, inflicting a mortal wound."