With your son as your dependent, you can file Head of Household. If you don't want to wait for your wife's Social Security number, then file now using the Head of Household status, and if it makes a difference in your overall tax amend to Married Filing Jointly after you receive the SSN for your wife.
Amended returns take a number of months to process, but if you're due an additional refund it will be sent once you file that return.
Otherwise, filing an extension and trying to wait it out is another acceptable route you can go.
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