the SSA does not have record of the name change until a person files for a name change. Given that many people do not change their names, or change it to something other than the spouses name, and even then, many states do not have any formal claim of a name change associated with the marriage and due to that, it is actually more of a common law name change associated with an event rather than a court ordered change, there is no way the SSA would even know what name you are taking, if anything different than your current name, until such time you filed for an updated name at the SSA.
so dogmatique;
had you applied for a new card with your new name on it yet?
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yeah!!!!!!!!!!
the answer to the OP is:
yes, you can. You may run into issues just as anybody that changes a name under common law would but yes, you can use whatever name you wish to use. It would be simpler to wait but you do not have to.

