Anti-Catholicism, perhaps?
It was definitely Anti-Catholicism. In Tennessee and Oklahoma, for example, Nixon got a higher percentage of the vote in 1960 than Eisenhower had in 1956. Oklahoma was in fact, Nixon's third-best state, after Nebraska and Kansas (it also held this title in 1972, being Nixon's third-best that year after Mississippi and Georgia). The same factor (Anti-Catholicism) was true in 1928; Hoover won the two states with percentages well above those of Coolidge (who actually lost them to Davis) in 1924, and also, above those of Harding in 1920. Oklahoma actually gave Hoover more than 60% of the vote that year against Al Smith.
I wonder if there’s any correlation between where Nixon 1960 performed better than Eisenhower 1952 or 1956 or Nixon 1968 and where McCain and/or Romney performed better than Bush 2000 or 2004.
Oklahoma and Tennessee are two states where this would obviously be the case. In 2008, Obama did worse in Tennessee than John Kerry. And in Oklahoma, Obama did better than Kerry in Oklahoma and Tulsa Counties, but significantly worse in "Little Dixie", the traditionally Democratic rural areas of Oklahoma. In 2012, Obama lost even more ground across both states, outside of their major metropolitan areas. In 1960, Nixon won several of the "Little Dixie" counties that Eisenhower had lost to Stevenson twice, exactly the same kinds of counties that swung heavily against Obama nearly half a century later. In 2008 and 2012, of course, racism and Islamophobia (remember the rumors about Obama being a secret Muslim?) replaced anti-Catholicism, but the overall factor of prejudice driving the electoral results was the same.