I think that whoever it is will still lose in a simelar fassion to Eisenhower, unless you have other pods (and even then it'll be a stretch).
I'm not overly sure who all the alternatives to Adligh are though, but it'd be interesting if they take either JFK or LBJ to be their running mate. If I understand it correctly, both Kennedy and Jonson were in the running for the second spot in 1956, a different nominee could easily result in either of them ending up as the running mate on the Dem Ticket. That could either boost their respective reputations for 1960 or harm them, depending on how they come across in 1956. If LBJ is nominated for Veep in 1956, I can't see him excepting the Vice Presidencial nod in 1960 as well, so even if JFK is still the 1960 nominee, he probably runs with someone else.
I've heard it said before that LBJ was pretty crucial to Kennedy's election and a different running mate would have resulted in president Nixon come January 20th 1961. Asuming JFK and his alternate VP still win, the US (and the world) could be in for an even more interesting 1960's.