I may be wrong, but I think Kennedy won both Carolinas, Georgia, Alabama, Arkansas, Louisiana, Texas, and West Virginia in the '60 election.
By 1960, conservative Southern Democrats had become increasingly wary of the liberal wing of their party, as represented by Adlai Stevenson. However, John Kennedy, an Irish Catholic from Boston, was not perceived as a liberal, especially in light of his father's conservatism and his refusal to condemn Joe McCarthy. His religion was a detriment in the South at that time, but more so in the Upper South, where there were few blacks and the white population was predominantly Scotch-Irish, with their historic animosity to the Irish Catholics. Race was more important an issue than religion in the Lower South, which also was home to the South's small white Catholic minority, from the port cities of Savannah and Charleston through South Louisiana to the German and Czech Catholic settlements of Central Texas.
By 1963, the region's image of Kennedy had changed. He was as willing as Eisenhower had been to enforce Brown vs. Topeka. His "best and brightest" bore many similarities to Roosevelt's "Brain Trust," a group of East Coast, Ivy League advisors that made Southerners uncomfortable. Unlike FDR, JFK made little effort to assauge the South and relegated LBJ to the outer fringes of his administration.
Had Kennedy not been killed, he would have found himself in an awkward position in 1964 as Martin Luther King and other civil rights leaders intensified public pressure for greater Federal protection of black access to public accomodations and the voting booth. Acceeding to black demands or holding the line would have jeopardized the political base that enabled him to gain a narrow victory in 1960. Kennedy just didn't have the political skills that Johnson had to get things done.