Your questions are very interesting, but tricky to answer at the same time. I will try to answer your question, but I apologize if it is not the exact answer your looking for. Part of the difficulty in answering your question is that we often assume that white people who supported the Civil Rights Movement, or the longer black freedom struggle, did so wholeheartedly. But historians have shown that white people from across the country and the political spectrum supported aspects of reform without supporting the broader push for racial equality. Being opposed to segregation did not necessarily mean one was antiracist or supported full racial equality.

One example of this would include Cherisse Jones-Branch's book Crossing the Line: Women's Interracial Activism in South Carolina during and after World War II. Jones-Branch makes that argument that white women were able to support some parts of black activism in South Carolina in the 1940s and 1950s because of their shared religious and class backgrounds with black women in organizations like the YWCA and Church Women United. The shared experiences of education and religious belief allowed white and black women to come together and push for reform, and they faced extreme backlash because of it. But this unity had its limits. Black women were rarely allowed to have leadership positions within these organizations, and white women actively opposed efforts by black women to have more say in the politics of said organizations. Many white women in these organizations also supported segregation, but they believed it should be made to be "separate, but equal" in reality rather than in word. White women were able to push for reforms they believed made segregation more equitable/humane while still upholding systems of racial inequality.

Eric Shickler also shows that Northern voters did not necessarily support all forms of Civil Rights reform between the New Deal and the Voting Rights Act in 1965. In his article "New Deal Liberalism and Racial Liberalism in the Mass Public, 1937-1968", Shickler shows that Northern voters overwhelmingly supported reforms that created legal racial equality, but opposed policies that called for racial integration. Although party differences became very apparent when it came to issues of anti-lynching and fair employment legislation, white voters across the board were unwilling to support housing reform. This would hold true into the 1970s when white flight and the busing crisis came to the front of national politics. Support for on-paper racial equality in employment and civil rights did not equate to support for physical integration.

While more conservative and moderate white people were able to support Civil Rights policies that created a more gradual change, or upheld segregation, small numbers of white people did support more wholesale legislation for racial equality. In Shickler's article, he points to the CIO (Congress of Industrial Organizations) as an important group pushing for racial equality. Home to some of the more radical labor unions in the US, including the United Mine Workers and the Longshoremen, the CIO was at the forefront of labor activism and were much more receptive to including black laborers--unlike the AFL. Part of the reason for the CIO's racial views was the sizeable presence of Communists and Socialists within its rank-and-file membership, as well as the leadership. Both Robin Kelley and Glenda Gilmore have shown that Communists, although a small group, were one of the few integrated groups at the forefront of the black freedom struggle. In his book Hammer and Hoe: Alabama Communists during the Great Depression, Kelley makes the argument that the Communist Party was one of the few spaces that offered black people leadership roles and a political voice during the 1930s. Some white people, were also involved in the organization, although the CP's mixed-race meetings made it the target of state repression. Gilmore also places the CP at the forefront of the black freedom struggle throughout the first half of the Twentieth Century in her book Defying Dixie: The Radical Roots of Civil Rights, 1919-1950. Similar to Kelley, Gilmore shows that labor activism and radical politics provided one of the few routes through which white and black people could fight for racial equality. Working-class identity and a radical commitment to equality for all people offered a foundation for white and black people to work side-by-side towards racial equality, although there is a lot more that can be said about that. This alliance was also just as fraught as the ones discussed earlier. Black people were still excluded from many labor unions, even in the CIO, and leadership struggles and racial prejudice plagued radical political groups like the Communist Party.

To summarize, there were ways for white people to fight against segregation and racism. Often interracial alliances were formed on the basis of other shared identities--whether class, gender, or religion. But support for certain reforms or legislation did not necessarily make a white person antiracist. Many white people supported reform of segregation while still believing black people to be inferior. It is certainly a complicated question, but hopefully this helps shed some light on the topic, and I hope that others can expand on some of the points I made here.