The next inauguration is both an endpoint and a restart. It closes one four-year term while beginning another, whether the same president has been reelected or a successor takes office.
The constitutional boundary
The Twentieth Amendment fixes the end of presidential and vice-presidential terms at noon on January 20. The outgoing president retains authority until that boundary; the incoming president does not exercise the office before it.
If the president is reelected
A second inauguration begins a new term without transferring the presidency to a different person. Personnel, policy, and administration may continue, but the political environment changes because the Twenty-Second Amendment generally prevents another presidential election victory.
If a successor takes office
Executive responsibility transfers to a new administration. Records, personnel, national-security information, emergency planning, and agency operations must move through an organized transition while the government continues functioning.
The clock begins again
POTUS Clock measures the regular constitutional term, not a prediction about political events. At noon, the count begins again at day one and the opening governing environment becomes the current stage of the presidency.