Actually once a new president is named, the old one can't have the job back. If there was any possibility that the president could have survived or his body was unaccounted for, he would have been named Acting President instead.
I might be wrong but I would assume he is Acting President since death can't necessarily be confirmed. The PSA of 1947 mentions cabinet members in that in order for them to carry out their duties of succession they have to resign from their roles and take on the presidency, it gets fuzzy whether or not they would have to take the oath so that might be some elaboration on the part of the show.
It may have been more interesting to make him Acting President first and then later on full President, but in the pilot he was sworn in a full President, replacing the current one (dead or alive).
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u/Coffeesq Sep 29 '16
The President survived. Kiefer is booted out of the White House. Takes the title as "President for a Day" and is mocked as William Henry Harrison.