Check out Greene county, Tennessee.Greene county is east of Knox and northeast of McMinn in Tennessee and not too far out of the neighborhood so to speak.
In 1850 census, I have a Sandy BROGLIN who was 22 and married to Tabitha who was 25.They had a son William A. who was 4 years old (b. c1846) and a son Tilman who was 2. (BROGLIN is one of the many misspellings of BROGD*N that I have found.)
By 1860 census, I think that "Sandy BROGLIN" had become Alexander BRODGON who was 31 and married to Elizabeth who was 18.They had a son William who was 13 and a son Tillman who was 12.
Though there are some differences between these two family records, I believe that this is a match for your William, particularly given that he named his first-born "William Alexander."
I am interested in the Brogd*n families in all three of these counties (Greene, Knox, and McMinn upto 1900) and in particular, I am interested in the family of any Brogd*n who has a son named "Tillman."
Please let me know if this information connects up your ancestor so that I can enter the data in my files.