"White Southerners still hated Black people" oversimplifies pretty drastically (like "Modern women hated men for decades"). What fraction of the white population feared their neighbours strongly enough to arouse such an active passion as hate?

" Perhaps you can explain..."
All the KKK or their ilk need, to persecute freely, is a background level of suspicion against certain people and a sense that these aren't part of "us." Bystanders holding these beliefs permit any ruthlessness, condone any horror. Callousness harms as much as or more than hate.