There had been quite a few major achievements in civil rights by 1960.
Some of these achievements had not really been achieved by “the movement” but were important nonetheless. The most important of these were the integration of the armed forces and the integration of Major League Baseball. Neither of these was caused by the Civil Rights Movement, which did not really exist at that time. However, both were very important at least in a symbolic way.
Other gains were achieved by the movement either in the courts or through direct action. The most important gain through the courts was the overturning of legalized segregation in schools. This came about in 1954 in Brown v. Board of Education. By 1960, the movement had also won a high-profile episode of integration in Little Rock. Through direct action, the most important achievement was the desegregation of public transit in Montgomery, Alabama. This was achieved through the famous Montgomery Bus Boycott which put Rosa Parks in the history books and brought Martin Luther King, Jr. to national prominence.
Although many important achievements happened after 1960, we should not ignore the significance of the movement’s gains before that time.