A lot of the feud dealt with one of the McCoys coming home from the war as a Union soldier and getting killed in early 1865, so if the war ends before that the feud might be less serious. Also the Hatfields were pro-Confederate, there could be incidents with partisanship on both sides of the border as the McCoys were pro-Union. Maybe also have the relationship between the Hatfield patriarch's son and Rosanna McCoy go more smoothly and avoid a subsequent murder. There will also be disputes more likely over mineral rights and coal in the region as time goes by. The trial that ended much of the feud would now also be an international incident in and of itself, and it might require the CSA and USA to permit extradition in such cases. This is not so far flung as it might appear especially a generation after the war is over but this area has *very* long memories...
If nothing else it might lead to the development of border patrol forces in the area or some sort of military outposts at Beckley WV and Pikeville KY respectively.