Only the marriage document (foreign or domestic) is needed to establish the name change if it meets the following criteria:
-- The new name can be derived from the marriage document, i.e. bride takes the groom's last name, groom takes bride's last name, compound name (with or without hyphen) of each spouse's original name for either or both parties.
Note: State law must also allow the type of name change wishing to be made
-- The marriage occurred within the past two years
-- The marriage document shows sufficient identifying information, e.g., biographical data
such as age, date of birth or parent's names, and
-- The identifying information matches the information on his/her current Social Security
number record.
The applicant will need to submit an identity document in the old name along with the marriage document if:
-- The marriage occurred within the last two years and has biographical information, e.g.,
age, date of birth or parent's names which does not match our records, e.g., the SSN
record.
-- The marriage occurred within the last two years, but does not show biographical
information, or
-- The marriage occurred over two years ago (with or without biographical information.)
The identity document used to establish the old name must match the name on the current SSN record. The identity document is acceptable even if over two years old or expired.
This is in addition to proof of current lawful employment authorized status.
You can refer SSA office personnel to EM-06064 for current procedures for a name change due to marriage.
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If you want to make other changes to the name on your SSN record other than what the state allows through marriage you will need a legal name change or have an immigration document issued to you in that name.
When issuing immigration documents, the Department of State and Department of Homeland Security issue them in the person’s legal name. The legal name is also generally the name in which the foreign passport was issued.
When an alien applies for an SSN card, SSA presumes the name on the immigration document is the legal name unless the applicant presents evidence of a legal name change (e.g., marriage) that occurred after the immigration document was issued.
In cases where an alien applies for a replacement SSN card and submits an immigration document showing a name that is different from the name on the prior Numident record, accept the immigration document as evidence of the legal name. In these cases, the prior SSN card may have been issued in a name that was not the NH’s legal name.
https://s044a90.ssa.gov/apps10/poms.nsf/lnx/0100203210RM 00203.210 Changing Numident Name Data