My great aunt’s book has a reference to the Klan’s response to violence by Black soldiers who killed a Confederate veteran in his home. I would add a book for you by Giles Vandal, “Rethinking Southern Violence- Homicides in Post-Civil War Louisiana, 1866-1884.” It has a lot of the detail you are looking for.

After the war, northerners came to view southerners with more sympathy and adopted their own form of racism towards “inferior Blacks.” The state of Delaware passed numerous acts restricting Negro suffrage. I do not have any organization names. Along with the Klan in the South, were the White Man’s Party (Alabama), Red Shirts (South Carolina), White Liners (Mississippi), White Leagers (Louisiana), and Knights of the White Camelia.