You can get moved up in the security line if your flight is due to board in under 30 minutes, but the new rules include a deadline by which your checked bags must be on their way to the hold.
Each airline's time requirement is different, but that deadline is iron-clad. Miss it and you don't get on the plane. This deadline cost us $1000 a couple of months ago, when we were in the check-in line in plenty of time, but did not actually reach the ticket counter until 58 minutes before the flight's scheduled departure. The deadline was one hour, and boy, did we pay for not knowing that. Also, be aware that the deadlines might be earlier for the skycaps than at the ticket counter, and make sure you know the details.
This was AA. We were told that our trip was cancelled as we had tickets that were not accepted for standby, and also non-refundable. DH wanted to buy one-way tickets on SWA to get where we were going, but I knew that would not work, if you don't show up for the outbound flight, the airline cancels your return tickets. Eventually, we talked them into selling us one-way outbound tickets on the next flight, which was nearly empty, BTW. They charged us $330 for each one of them.
The moral is, if you are running at all late, get someone in line with those bags first thing, before you worry about something like returning the rental car.