Yes, once the spouse has an SSN, they should use it to file their return.
If they were married and living together in 2021-2023, he should not have filed head of household; the only statuses legally available to him were married filing jointly or married filing separately. Those returns can be amended to correct the filing status to married filing jointly. The 2021 amended return needs to be filed no later than April 15, 2025.
If the spouse had income in the prior years, her income needs to be added to the amended returns.
See this IRS webpage for more information on filing status.
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