This is the line that is used later on in the query to retrieve the events:
$_today = new \IPS\calendar\Date( "now", \IPS\Member::loggedIn()->timezone ? new \DateTimeZone( \IPS\Member::loggedIn()->timezone ) : NULL );
Changing the "now" to "today" and you'll get those already started events to appear - but that's a direct code edit.
I think I might be able to do a plugin to help you with this though...gimme a few minutes to check.
Here you go - try this out:
(NE) Upcoming Events - show recently started events.xml