If we are going to look at history, let's go back to the very first impeachment trial of Blount
https://constitutioncenter.org/blog/...d-and-expelled
I'll skip to the relevant part.
During Blount’s absentee proceedings in early 1799, the Senate didn’t move forward with a trial against Blount, deciding that he had already been expelled from the Senate. By a 14-11 vote, a resolution was defeated that read that “William Blount was a civil officer of the United States and therefore liable to impeachment.”
So the very first time the Senate had the chance to convict someone that had already been expelled, they punted. No reason the Senate couldn't do the same thing. A trial is not mandatory apparently.
Originally Posted by HedonistForever
And subsequent to that Belknap was impeached and tried even after he resigned in an attempt to prevent his impeachment
"On March 2, 1876, just minutes before the House of Representatives was scheduled to vote on articles of impeachment, Belknap raced to the White House, handed Grant his resignation, and burst into tears.
This failed to stop the House. Later that day, members voted unanimously to send the Senate five articles of impeachment, charging Belknap with “criminally disregarding his duty as Secretary of War and basely prostituting his high office to his lust for private gain.”
The Senate convened its trial in early April, with Belknap present, after agreeing that it retained impeachment jurisdiction over former government officials. During May, the Senate heard more than 40 witnesses, as House managers argued that Belknap should not be allowed to escape from justice simply by resigning his office.
On August 1, 1876, the Senate rendered a majority vote against Belknap on all five articles. As each vote fell short of the necessary two-thirds, however, he won acquittal. Belknap was not prosecuted further; he died in 1890."
But you knew this already and only provided what you thought helped your point. As I have noted time and again, the Senate has precedent to try an ex-official, which an ex-president is.