There's tons of interesting information about the history of American public opinion on this site which has some tables made from the American National Election Study dataset. Here we see answers to the question "Are you in favor of desegregation, strict segregation, or something in between." I really couldn't say what that "something in between" would be, but it sure seems to be a popular answer. Between 1964 and 1968, the rise in support for the "in between" position seems mostly driven by the decline in support for segregation, which is all to the good. Then in 1970, support for desegregation reaches its peak as both segregation and desegregation decline.
But then in the next three surveys, support for desegregation consistently declines as more and more people shift into this mysterious middle ground position. It's too bad that they stopped asking this question in 1978 because public opinion was shifting throughout the 1970s and not just in the predictable sense of a steady decline in support for segregation.