I had something similar happen last weekend. My DCA-ATL flight was delayed, and it looked like I was going to miss my connection. I called Delta and selected the callback option. At the airport, I cleared security and went to the customer service desk. The agent was able to book me on an earlier DCA-ATL flight (preserving my upgrade), and protect me on the first flight the next morning since I was scheduled to catch the last flight of the night out of ATL. She printed a receipt showing my confirmed reservations for DCA-ATL and both that night and the next morning's ATL-VPS flights.

Right after finishing with the agent, I received a callback from Delta. I told the phone CSR that I was pretty sure everything was resolved. I asked her to double-check that I was still confirmed for that night and protected for the morning. She told me I'd been removed from the evening flight and had to pick one or the other, so if I wanted to try to get the evening flight I'd give up my confirmed seat in the morning. I went back through the line for the desk, and the original agent there was able to get me back on both flights. As it turned out, I missed my connection and spent the night in ATL, so it wouldn't have mattered after all.