In 1960, Kennedy managed to gain many southern states, thanks most likely to Lyndon Johnson (certainly in Texas). However, he lost many as well. Virginia, Kentucky, Tennessee and Florida (home state of his friend, Senator George Smathers) went for Nixon. The electors in Mississippi and Alabama (the latter not totally but mostly) cast their votes for segregationist Democrat Harry F. Byrd in protest of the pro Civil Rights platform of the party. One faithless elector in Oklahoma did likewise.
The challenge here is to win Kennedy more of the South in the 1960 election.