One possibility I hadn't thought of.
Joshua Lawrence Chamberlain was Governor of Maine, so clearly happy to go into politics. As a Union general, he had distinguished himself at Gettysburg. Would be an obvious "counter" to Hancock should the latter be nominated, and a perfectly credible one against any Dem.
Slogan. "He saved Little Round Top. Now let him save the nation!"
Chamberlain also probably fits the bill of not being soft on reconstruction, which Sherman definitely would not. IMO it's him or Sharridan, if not Grant.
The real question is who becomes General-in-chief if Grant dies or has a break-down? Grant and Stanton were both very sympathetic to reconstruction. Sherman, Grant's logical successor, was not. This could make congressional Republicans even more anti-Johnson. So you have to step back and reflect on the fact that, absent Grant, there's a non-tryvial chance Johnson gets impeached. So to avoid that butterfly, you'd need Grant to be incapacitated as late as possible.
Assuming it's Chamberlain--and that seems to me to be the most likely outcome in 68--the question is, who become his key advisors? I think we can assume the Conkling machine won't play as pivotal a role in his admin as they did in Grant's. Who are his political backers? Blaine? Sumner? Other New England politicians? How does he handle issues of foreign policy?
Could be interesting times.