If you show up at the airport inside some window (2 hours?) of when you were originally scheduled to leave, you can stand by for the next flight with no penalty on any type of ticket.
I was in this situation this past spring. I called the AA Executive Desk. They told me the next flight was empty, that they could confirm me on it for a $100 change fee, or that I could show up late and get on it for no extra charge. They didn't quite say I'd be crazy to pay - their calls are recorded and management reviews some of them - but the message was clear. Wby pay when you don't have to?
Beyond that window, it probably makes sense to rebook, though if the flights are empty (most are these days) showing up at the airport and throwing yourself on their mercy might work. Worst case, you'll pay the change fee then rather than earlier. I'd still call ahead to make sure that they won't cancel your ticket completely. I don't think they would, but lots of things are different right now.