Symington is the most likely alternative. I think JFK will still win. LBJ's refusal may cost JFK Texas, but even if it costs him that state and South Carolina (which next to Texas was JFK's narrowest southern victory), JFK would still get 271 electoral votes--more than enough to win.
http://psephos.adam-carr.net/countries/u/usa/pres/1960.txt And even if JFK loses North Carolina as well (52.1-47.9 for JFK in OTL) that doesn't mean Nixon wins; rather, the race goes into the House (with each delegation having one vote) where JFK has the advantage, as I explain at
https://groups.google.com/d/msg/soc.history.what-if/GgIKHDCwhEo/7W1UrFH6cDQJ...
The one scenario in which not having LBJ on the ticket would elect Nixon is if Nixon carries not only Texas, North Carolina, and South Carolina, but Louisiana as well. The reason this is (just barely) plausible is that while JFK won it pretty comfortably in OTL--by 50.4 percent to 28.6 percent for Nixon and the rest for unpledged electors--the state Democratic committee only narrowly voted to back JFK over unpledged (presumably pro-Byrd) electors. If not for the choice of LBJ, it might have chosen the latter, so that JFK would in effect be a "third party" candidate in Louisiana, the Byrd forces having the official Democratic designation. But even then I don't see how Nixon, who only got 28.6 percent of the vote in the state in OTL can carry the state. And for Louisiana to go to the unpledged electors will not make any difference if the race is going into the House anyway. (Moreover, even without official party support, JFK would still have a good chance to carry Louisiana in a three-way race, because unlike most other southern states, there is a substantial Catholic vote.)