Can anyone give me suggestions for how to track down an illegitimate birth in 1880 in Oconee County?

I am specifically interested in the Jan 1880 birth of a daughter to Mary E. Tate, who is found in the 1880 census living with her mother Elizabeth Tate in Oconee County, Tugaloo District. It is unlcear whether the infant grandchild listed in the census is the daughter of Mary, or of her sister Amanda.

This illegitimate daughter is my great grandmother, and I have hit a roadblock.

Thank you very much.