On the very last day of the 30-day pinger battery window
No such thing. It's been explained they are designed to generate pings with a defined signal strength for at least 30 days... but they don't suddenly stop at midnight on day 30. They have said they typically last more like 40 days and at gradually reducing signal strength.
As I recall they justified listening for the pinger by saying that "additional analysis" of sat data had narrowed down the search area to make it worth a shot. The details of that additional analysis haven't been released as far as I can tell.