For the record, the electoral vote count in 1960 was:
Illinois, 27
Texas, 24
South Carolina, 8
New Mexico, 4
Hawaii, 3
The electoral vote totals were 303 for Kennedy, 219 for Nixon, and 15 for Harry F. Byrd (one of those was a Republican who should have voted for Nixon). 269 votes were needed to win (the total was 537--the House had 437 members because Alaska and Hawaii each had one Congressman, having been admitted after the 1950 census...the 3 votes for DC were first cast in 1964).
IL & TX alone were enough to flip it, the rest were superfluous.
I never heard of any fraud in SC on behalf of Kennedy, and it's doubtful the Democrats running the state at the time were going to bother giving him a boost (Sen. Thurmond surely wasn't).
Hawaii in 1960 was still a semi-GOP state (albeit in transition to 'Rat since the mid '50s), and how much fraud that went on there is questionable, and there was a GOP Governor at the time.
As for NM, just a bit of fraud was needed in the southern tier counties along with heavily Hispanic northern ones (like Rio Arriba, which has had corruption for years) to flip the state, although a Republican (and former Governor, Ed Mechem) did beat the incumbent Dem Governor in that election, so one might wonder how come fraud didn't help John Burroughs win a 2nd term.