The two states have waged an unusual court battle over the agent facing charges over a shooting in Minneapolis. The agent could be released from jail on Thursday.
The order came amid a spate of legal challenges, including one by Democratic state attorneys general, that could still result in the rules being blocked.
Under its new plan, the Postal Service would not deliver mail ballots in states that do not share voter data with the agency and would deliver only ballots that meet certain criteria.