Best chance would probably be to have the more precise order not get to him at Shiloh - he led his men to a point near where they were behind CSA lines because of miscommunication, and had Grant not gotten a more precise order, or something like that, to him, he could attack.
So, let's have him attack but get injured(letting him write Ben Hur while recovering). He is credited with being part of Grant's surprise strategy, the bigger win knocks the CSA down a bit faster, so Vicksburg and then whole CSA each fall a few months earlier than OTL.
he might become a good candidate to follow Grant in 1876(Hayes wasn't chosen on the first ballot OTL, so with a more well-known name, he could win). Perhaps he has been in the military a bit after the war OTL to be part of that negotiating, too, which gives him even more notoriety.
He seems like he would be an interesting fellow, but I really don't have much knowledge of his plolicies. However, it appears this is a very plausible route for him if he were to be part of a huge Union win at Shiloh, rather than the bloodbath victory it was OTL.