There's no clear legal precedent establishing that someone can be impeached, convicted, and punished after leaving office. It's been attempted, but never completed.
There's precedent for a trial after the person has left office, but since he was acquitted, there's no way to know whether a conviction would have been upheld by the courts.
The question is jurisdiction. The House vote was unanimous. The Senate found it had jurisdiction, the accused admitted the Senate had jurisdiction, and the trial was held.
At the end of the trial, the vote is on conviction or acquittal, not jurisdiction. The vote was a majority for conviction, but less than the two-thirds required.
There is no appeal of an impeachment to the courts. The Constitution gives sole authority over impeachment trials to the Senate.
There's no clear legal precedent establishing that someone can be impeached, convicted, and punished after leaving office. It's been attempted, but never completed.
There is clear legal precedent that the House may impeach, and the Senate has jurisdiction to proceed to hold a trial after the accused has left office. Where they have jurisdiction to try the accused and return a verdict, that verdict may be guilty.
The question of jurisdiction is argued pre-trial. Where a lack of jurisdiction is found, the trial does not proceed. If more than a third of the senators later refuse to convict because they object to the finding of jurisdiction, there is an acquittal on something that resembles jury nullification.
https://en.wikipedia.org/wiki/William_W._Belknap
Starting on April 5, 1876, Belknap was tried by the Senate. For several weeks Senators argued over whether the Senate had jurisdiction to put Belknap on trial since he had already resigned office in March. Belknap's defense managers argued that the Senate had no jurisdiction; the Senate ruled by a vote of 37–29 that it did.
Belknap argued that the vote on jurisdiction was less than two-thirds. Two-thirds is the standard required on a vote for conviction, not a procedural vote. The Senate went ahead with the trial.
The courts have no say in impeachments. The Senators alone make up the rules. They found jurisdiction with a majority vote of less than two-thirds. The current crop of senators can do it again.