The records show a John O. Smith who enlisted in Co. "I", 17th Alabama Infantry Regiment, Aug. 21, 1862. He gives his place of residence as Montgomery County, but his age doesn't match the birthdate you provided. Do you happen to know where the man you're attempting to identify lived in 1860? That would be helpful.

I don't see another John O. or J. O. Smith in an Alabama command. Of course it's possible he enlisted simply as John Smith, or that the "O" initial was mistaken for an "A".