You already know about a couple of men named A. J. Conner who facts don't fit your ancestry. It's possible that we haven't explored other men by that named who enlisted in military commands from other states. I hoped to find a Connor in a Butler County company that served as 2nd Co. "H", 1st Georgia Confederate Regt., but was disappointed.
It's also possible for a man with a late enlistment date that no records survive. I have a direct ancestor for whom there is no evidence of military service. Born in 1820, David Henry Ellard lost his first wife in 1857. He remarried in May 1863, and my gr. grandfather was born on April 23, 1865. DHE never returned from military service, the family story being that he died at Talladega as the war ended, probably April or May 1865.
His mother named the boy for his father, David Henry Ellard. DHE could have been enrolled in a militia command organized during the summer of 1864, or been conscripted or simply been a civil employee of a company having a contract with the Confederate government. I dont know, and don't ever expect to learn what really happened to my ancestor. DHE's oldest son James Ellard enrolled during the winter of 1863-64 and served in Co. "H", 24th Alabama Regiment. He never returned home, either.
His widow remarried and my gr grandfather became an unwanted step child. That must have been a common experience following the war. Of course remariage means there would have been no pension application by the widow. DHE Jr lived until 1947 and related what he knew about his father to his many daughters. Only one son named Shelby lived to adulthood. He always said that that having all those girls was a poor farmer's luck.
The place my family lived along the Perry-Bibb County line is still called Ellard's School. There's also an Ellard Creek nearby. My father was born near Heiberger, another geographical expression along the road from Brent to Marion AL.
http://www.bing.com/maps/default.aspx?encType=1&where1=Ellards%2c+Alabama&cp=32.814289~-87.261284&qpvt=Ellards+Alabama+map&FORM=MIRE