As some of the posts in this thread have alluded, it's really more of a mystery why the state voted so differently than the other Deep South states, where Stevenson won by double digits (2 to 1 in Georgia!). It voted more like its neighbor to the west did...
Louisiana in '80 also gave Reagan a bigger margin than any other Deep South states.
In both cases, the difference is substantially the French Catholics, who voted heavily for Eisenhower in 1956 as a result of his targetting the Catholic vote, and were unhappy with Carter’s record on abortion and contraception in 1980.
Additionally, many anti-Long Democrats even outside Acadiana voted Republican for President in 1956, and “Calthrina950”
has said about the Long northern upcountry parishes where Stevenson did very poorly compared to other regions of the Deep South upcountry:
...I've noticed that those same [[northern upcountry] parishes largely split between Nixon and Unpledged Electors in 1960 as well, and in 1964, Goldwater dominated the region against Lyndon Johnson. LaSalle Parish, in particular, gave Eisenhower more than 60% of the vote, and it was one of the parishes to give Nixon a majority four years afterwards [1960]. I have read elsewhere, though, that Northern Louisiana is notorious for its racism, and this certainly played a factor in its voting patterns (obviously). The racial tensions in that region must have been more pervasive compared to analogous regions in Northern Alabama, Northern Georgia, Northern South Carolina, and even Northern Mississippi—where Stevenson, Kennedy, and Johnson held up in support despite civil rights.
