Yes, segregation of "whites" from "other races" did occur. For example the Virginia Racial Integrity Act required that marriage certificates record the race of persons as "[...]Caucasian, Negro, Mongolian, American Indian, Asiatic Indian, Malay[...]" and forbade "[a] white person in this State to marry any save a white person, or a person with no other admixture of blood than white and American Indian"

In 1927 the Supreme Court ruled in Lum vs Rice that a child of Chinese descent was "[a] member of the Mongolian or yellow race, and [275 U.S. 78, 82] therefore not entitled to attend the schools provided by law in the state of Mississippi for children of the white or Caucasian race."

Outside of the South other states also had segregation. For example California's segregation of children of Mexican descent in the education system was overturned in Mendez v. Westminster.

In different times and different places a person regarded as 'Coloured' or 'White'. In 1954, Hernandez vs Texas, the state claimed that ww2 veteran Hernandez hadn't been discriminated against as he was not black. The Supreme Court ruled his 14th Amendment rights had been violated.

Partly Colored: Asian Americans and Racial Anomaly in the Segregated South by Leslie Bow was recommended to me on this subject, but I haven't read it yet.