After Reconstruction is a bit trickier, but it can be done. The South is a very statist-oriented region, be in it's support of state-led economic development or in it's support of state-sponsored 'family values' and the like, so I think it's absolutely doable.
Here's my POD: The Dyer Anti-Lynching Bill is passed into law in 1922.
This gives the federal government the ability to prosecute lynchings, leading to a noticeable decline in the activity in the otherwise racially charged twenties. Rolling on this success, President Coolidge and the Congress passes a Civil Rights Act in 1925, essentially an earlier version of the 1957 act, guaranteeing African-Americans some semblance of voting rights.
Thus, the Civil Rights movement starts a bit earlier than IOTL, and in 1928, Coolidge and the Congress pass another Civil Rights act, essentially the one passed in 1960, but again, earlier. The end of Coolidge's term sees increased racial violence in the South, but otherwise, the country seems to be on the right track. Herbert Hoover defeats Al Smith thanks to anti-Catholic bigotry combined with economic prosperity, and the Hoover administration promises to continue the civil rights work of its predecessors.
With the Great Crash in 1929 and the subsequent collapse of the Hoover presidency, the Democrats take control of Congress in 1930, making further civil rights legislation look dead in the water. However, Hoover cuts deals with northern Democrats and the shell of the Republican Party that still exists in the House and Senate to overcome Senate Democrats' opposition to the passage of a Civil Rights Act in 1932, which itself essentially did the handiwork of the Civil Rights Act of 1964 two decades earlier.
Hoover is out of office in March, and President Roosevelt comes into office non-committal on civil rights issues, as not to arouse his southern base and destroy his party. However, with Southern Democrats having no other outlet, with the Republicans having stayed firmly in support of civil rights, Roosevelt gambles and decides to use the nascent FBI, under the leadership of Melvin Purvis, to enforce the laws on the books and allow African-Americans the right to vote.
As a sop to the South, however, Roosevelt does invest more heavily in bringing all southerners, black and white, into the 20th Century economy. Roosevelt wins re-election in 1936 with a majority of the African-American vote, the first Democrat to do so. With the passage of Richard Wagner's hospital insurance bill in 1938, both black and white southerners are given access to health insurance, and with the work of the WPA and other New Deal agencies in strategic building and public information campaigns against racism, the effects of racial prejudice begin to wane in the South, and a biracial coalition becomes an option in the near future for the Democrats.
World War II further reduces the appeal of racism, with the Roosevelt administration making a conceited effort to reinforce the idea that racism was a Nazi trait, not an American one. Roosevelt was re-elected in 1940 and 1944 thanks to a solid majority in the South of both white Democrats and to a lesser extent, black Democrats, though the latter were far more prevalent in the north.
By 2010, the 'solid south' was still a reality, though for different reasons. The region, having become more affluent, had contributed a significant block of its voters to the neoliberal Republican Party, though the same basic fault lines existed as they did in 1940. The Democrats, now composed of a strong majority of middle class and lower class whites and blacks, can usually count on winning a good portion of the South every election cycle.
