Surname from DNA

My maternal great grandmother called herself Rosa Schmidt, but she was actually of Lithuanian Jewish descent according to DNA from my mother. Is there any way to match Jewish DNA to Jewish Lithuanian surnames?

Rosa Schmidt had a colorful past and there must have been a reason why she hid her identity. She came to the U.S. from Swilno, Russland in 1892. According to my aunt, she was a prostitute in Cleveland when she met my grandmother’s father, a one-legged night watchman. She settled with him in Chicago, bringing with her a daughter. Rose had 4 more children before she died in 1916.