Often located in church Sunday school classrooms, these private schools replaced the segregated public schools and thus education remained mostly segregated. Since you've not heard much about them in recent decades you may have presumed they vanished. They did not.
Late in our respective academic careers, the other DrC interviewed for an administrative job at a large state university in the Deep South. While there, she was taken to visit some local public schools where she would place teacher candidates for student teaching.
Even in the university town virtually all of the public school students were BIPOC. It was an eye-opener. She had the same admin job in California at the time, and classes in CA were thoroughly integrated and multi-racial.
She asked and was told the white children were in "private or church schools." At the time home schooling wasn't yet common.
Several years later in retirement, we spent a year as visiting faculty in the greater Dallas area. Private schools were prevalent there, too. And a current news story makes clear that movement remains very much alive in today's Texas.