Since you probably won't have access to a birth record, consider some of these ideas.
Look for the younger sister, or any other siblings at the same time as any life changing events in their lives may name your subject.
1. Use Google to search marriages, obituaries, etc.
2. Look for a high school year book either at the local school or library.
3. Look for a marriage record at the county courthouse, say from 1955 forward. Using the last name may be difficult, but the records may list parents, and you might find something to look up in the newspaper, or Google.
4. Look for a death record. Same issues as #3.
5. Look for an obituary of your subject. Google or local library.
5. Look for obituaries of parents. Father first as mother may have remarried. Again, try Google or local library.
If you're not from the town where you're searching, call the local library to see if they can help you. Many libraries have local history rooms, and they may have an obituary index. Someone there will have local knowledge.