Social Security changes it legally then she has to literally change it everywhere else... DMV, the utility companies, your mortgage company, all your bills, bank accounts etc... They will want proof of the name change. Some ask for a copy of the social security letter /card and some will ask for the filed marriage license (with the stamp on it). The IRS is the only one that will just check the social security number on their own.
Assembly street...... social security building
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