I have checked the book THE REASON FOR THE TEARS by Bobby L. Lindsey and on page 234 with the heading "Fredonia" is this paragraph:
"Always progressive in the field of education, this place received it crowning laurels when the State Legislature established 'The Southern Military Academy' at the village. Financed by lottery, the state institution operated only a few years when forced to close by the civil strife. Colonel Gibson Hill and his wife, Selina, owned and occupied the academy buildings after the war."
This is the only reference I have found to The Southern Military Academy in the this book.