Does the Senate have jurisdiction to hold an a impeachment trial after Trump (or any other official) leaves office? Maybe yes, maybe no. See pp. 47-48 of this CRS report, discussing Sec. of War Belknap's post-resignation trial.
https://crsreports.congress.gov/product/pdf/R/R46013

There are two sections of the Constitution at play. Article 2, section 4, says:
The President, Vice President and all civil officers of the United States, shall be removed from office on impeachment for, and conviction of, treason, bribery, or other high crimes and misdemeanors.

So Article 2, Section 4 doesn't seem to deal with former officials. It deals with officials, who can be "removed." So let's look at Article I.

Article 1, Section 3, Clauses 6 and 7 say:

The Senate shall have the sole Power to try all Impeachments. When sitting for that Purpose, they shall be on Oath or Affirmation. When the President of the United States is tried ...

... the Chief Justice shall preside; And no Person shall be convicted without the Concurrence of two thirds of the Members present. ... Here's where is gets interesting ...

Judgement in Cases of Impreachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States: ...

... but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgement and Punishment, according to Law.

On the one hand, there are only references to "officials" and the "President," etc. Nothing said about former officials. Would the Chief Justice preside over a hypothetical impeachment trial of a FORMER President? That is not expressly addressed in the text of the Constitution.

Article 2, Section 4 says: The President, Vice President and all Civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.
Nothing THERE on former officials.

So the question seems to be whether the REMEDY: "Judgement in Cases of Impreachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States" implies jurisdiction ....

... to try a FORMER official solely for the purpose of disqualification, even though "removal" is no longer possible, and even though criminal prosecution actually is possible (in theory, depending on the facts).

My personal textual interpretation is that no jurisdiction exists to hold impeachment trials of former officials (nor future officials, nor mere candidates). BUT, to answer that question would require @marklevinshow-style knowledge of the Federalist Papers, English history, etc.

And whether or not such jurisdiction exists has absolutely nothing to do with Trump or how you feel about him (or Pelosi). It is a constitutional question for the ages.

Franklin argued that impeachment was needed because otherwise the only avenue to get rid of a rouge executive would be violence (unwanted by anyone). That logic does not apply to former officials.
https://webcache.googleusercontent.com/search?q=cache:PU4jxz8u4xQJ:https://www.smithsonianmag.com/smart-news/american-presidents-can-be-impeached-because-benjamin-franklin-thought-it-was-better-assassination-180961500/+&cd=1&hl=en&ct=clnk&gl=us