The South would revolt either with unpledged electors or a third party and Stevenson might only win Massachusetts if state pride is strong enough. Kennedy wouldn't even have his vote against the Civil Rights Act of 1957 yet to point to for them.
More Southern delegates voted for JFK than for Kefauver in the 1956 VP nomination fight. That's because most Southern Democratic pols HATED Kefauver and viewed him as a "Scalawag". They would be hard-pressed to do that if their delegations had more than done their part to put JFK over the top at the convention.
JFK actually took the lead on the 2nd ballot, but a number of Favorite Sons shifted their voted to Kefauver to give him the nod. According to Len O'Connor, the author of CLOUT: Mayor Daley and His City, Joe Kennedy got on the phone to chew out Mayor Daley for pushing JFK so effectively for VP. Papa Joe didn't want Jack tied to a losing ticket that was going down badly to a popular Eisenhower; the name of the 1956 game was to give young Jack exposure, not the nomination.