Dear Ms. Tate,
Thank you for posting your request on History Hub!
We suggest that you submit a Name Index Search to the US Customs and Immigration Service (USCIS) to obtain a copy of the files in their custody. When you submit an Index Search Request, the USCIS Genealogy Program will send you a letter reporting the search results. The letter will identify all file references found in the index and provide instructions and additional information necessary to request the file(s) from the USCIS Genealogy Program and the Freedom of Information (FOIA) Program.
We suggest that you contact the Tulare County Superior Court to inquire if any legal adoption papers were filed regarding your mother. Please review the FamilySearch Research wiki for California Adoption Research for more information.
Your mother may have obtained her card through a process similar to the current “Enumeration at Entry (EaE)” which allows noncitizens admitted for permanent residence to obtain SSNs and Social Security cards based on data collected as part of the immigration process. Please see Social Security Numbers for Noncitizens (EN-05-10096) and Social security numbers - Code of Federal Regulations for more information.
Lastly, we suggest that you contact the Social Security Administration for information on how to correct your mother’s name on her social security card. The naturalization information that you receive from USCIS should provide documentation.
We hope this is helpful. Best of luck with your family research!