Disclaimer: I made a mess of this answer because I had 3 hours sleep last night. I will return to it tomorrow to add to it, amend it, edit it, etc. There is some interesting content in there though, so I'll post what I have so far.

Great question! I'll try to give an answer, although I don’t think I can adequately answer your question in a single reddit comment. There have been books written dedicated to this entire aspect of jazz. Stanley Crouch wrote "Jazz predicted the civil rights movement more than any other art in America" in 2008. What does he mean by that? Well, he says it himself, jazz music is inherently democratic and not only that, the players are judged on how they play and how they play alone. So, I'd like to point out that jazz music didn't really have a Civil Rights leader in the same way that MLK was one, they (jazz musicians) often reluctantly adopted that role, jazz musicians became a collective symbol in the Civil Rights movement rather than any individual paving their way alone, although some became more symbolic than others. Anyway let's explore the impact jazz had on the Civil Rights movement.

Take a look at some different figures and see where that leads us:

Louis Armstrong: After the 1957 Little Rock Crisis, when the National Guard prevented nine black students from entering a high school, Armstrong cancelled a tour to the Soviet Union, and said publicly, “the way they’re treating my people in the South, the government can go to hell.” 6 days later Eisenhower ordered federal troops to escort students into the school. ^1

That outspoken attitude regarding segregation is laudable, but many jazz musicians criticized Armstrong for playing in front of segregated audiences and for not taking a strong enough stand in the civil rights movement and he was often labelled an "Uncle Tom" by his peers. ^2

Billie Holiday: What's interesting about Billie Holiday's contribution here isn't just the song Strange Fruit (written by a Jewish New Yorker named Lewis Allen who was married to a black woman named Anne Meerepol), it's how the performances in their early days were planned. The song was first performed by Holiday in the Cafe Society in Greenwich Village in 1939. Barney Josephson, founder of the society, arranged for the following: Holiday would close with it; the waiters would stop all service in advance; the room would be in darkness except for a spotlight on Holiday's face; and there would be no encore. ^3 The magnitude of what they were doing was not lost on Josephson, in fact he sought to amplify it, he literally shone a spotlight on the issue Holiday was singing about (lynching). In October 1939, Samuel Grafton of The New York Post described Strange Fruit: "If the anger of the exploited ever mounts high enough in the South, it now has its Marseillaise”. ^4 Lady Day’s impact on the Civil Rights movement should not be downplayed.

Max Roach: A great bebop drummer, he recorded the song We Insist, Freedom Now Suite in 1960 after being invited to contribute to commemorations of the 100th anniversary of the Emancipation Proclamation. He was also outspoken on Civil Rights and spent much of his time lecturing on social justice issues. (I can’t find sources for this one, but I’ll edit them in when I stumble upon them).

Charlie Mingus: Like Armstrong, Mingus’s reaction to the Rock Nine incident is the most famous regarding the Civil Rights movement. He penned Fables of Faubus, the lyrics offering some of the most blatant critiques of Jim Crow attitudes. Read the lyrics here. Mingus was a major figure in jazz at the time and his influence was far reaching. ^5

John Coltrane: Coltrane was drawn in to the Civil Rights movement in 1963, Coltrane played eight benefit concerts in support of Dr. King. He wrote a number of songs dedicated to the cause, but his song “Alabama” became an important part of the Civil Rights movement, the notes and phrasing of Coltrane’s lines are based on the words Martin Luther King spoke at the memorial service for the girls who died in the Birmingham bombing^6

Now, there are many key figures I omitted from this comment (Duke Ellington, Miles Davis, Nina Simone). There is simply no way I can answer your question adequately. Rather than focus on the individuals, even though many of them made major contributions to the Civil Rights movement, one should see the jazz movement as a musical movement that became politicised and interwoven with the Civil Rights movement. Recommended reading: The History of Jazz by Ted Gioia: “It is impossible to comprehend the free jazz movement of these same years without understanding how it fed on this powerful cultural shift in American society.” The book is an excellent read, very well written. It focuses on the music itself mainly, but doesn’t shy away from the social history that intertwines with the history of jazz.

And this gives more detail as well.

^1 "Louis Armstrong, Barring Soviet Tour, Denounces Eisenhower and Gov. Faubus". New York Times. September 19, 1957. Retrieved August 30, 2007. See also, from September

^2 Collier, James Lincoln (1985). Louis Armstrong. Pan Books. pp. 317-320.

^3 Lynskey, Dorian (2011). "33 Revolutions Per Minute". Faber & Faber Ltd.

^4 "Strange Fruit: the first great protest song". The Guardian. February 15, 2011.

^5 Jazz: Beneath the Underdog". Time. October 2, 1964.

^6 John Coltrane Biography". The John Coltrane Foundation. May 11, 2007