As has been pointed out already, not only is Hoover highly unlikely to beat both of these men in the 1960 election, but he's also unlikely to seek out the Presidency period (assuming he follows the same career trajectory as IOTL). If you want to see Hoover take the commander-in-chief spot, I think it'd be more likely in the event of an extreme terror attack, or some kind of Constitutional crisis. He wasn't a man who was interested in public opinion (or really anyone's opinions but his own), but he was certainly a man of action, and if faced with a situation where his country demanded a strongman to act, I have no doubt he'd fill that void. With that having been said, any situation with Hoover in command is going to be nasty. He'll make Nixon's shenanigans look like child's play, quite frankly. Suspension of civil rights, secret police (which he had effectively turned the FBI into anyways), martial law, etc.