African-Americans, who could only join the Democratic Party from 1924 onward, started leaving the Republican Party as early as 1936 not 1960 as is often believed. In fact in the election that year 71% of the Black vote went to FDR. This appears to be largely a consequence of the New Deal although the tireless campaigning of Eleanor Roosevelt may have played a role.
The voting trend held firm over the next three elections with African-Americans voting by a margin of over 3 to 1 in favour of Harry Truman over Thomas Dewey in the 1948 election. Truman of course was a key figure in ending racial discrimination in federal hiring often running afoul of key segments of his own party on this issue.
Even Eisenhower who would fully integrate the army (a process started by Truman but rejected by FDR) and opened up the schools to Black students was regularly outvoted at the polls by a margin of 76% to 24% in 1952 and 61% to 39% in 1956, with respect to the black voting demographic. However his appeal among African-American voters was a lot better than the last four Republican Party candidates (Alf Landon, Wendell Wilkie and Thomas Dewey- twice)