I found only one possibility in my database. Alas, he was found on the 1880 census already in Texas (Collin County), after which I've no data on him at all. With kin back in TN, it is not beyond the realm of possibility that William went back there when he was 19 or 20, met and married Rachel, and settled there briefly. Alas, her given name was very common amongst girls in TN cir 1870-1880, and even a restrictive search turned up too many for a closer narrowing without some location information.
If you've traced William's family forward and found a living male Abernathy of direct descent, we have Y-DNA data available for comparison from an Abernathy who descends from a brother of the Abernathy possibility I found in my database. I believe I can say without hesitation that he would be overjoyed to find and prove a new cousin from his sparsely populated line.
If William's line dwindled to females only, there's not much that can be proven at this point in time without, perhaps, a great deal of luck and a lot of tedious, time consuming research.