https://en.wikipedia.org/wiki/Twentieth_Amendment_to_the_United_States_Constitution#Text says the president's term begins/ends "at noon on the 20th day of January," but does not specify a timezone. It's accepted that this means eastern time, but I like to imagine that for a few hours the east and west of the US have different presidents.
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If you want to get really silly: https://en.wikipedia.org/wiki/Leap_second#Insertion_of_leap_seconds have extended a few presidential terms.