I did the similar thing at Heathrow Airport. I took the underground from London to Heathrow, but I only had Zones 1 & 2 travel card, while Heathrow is in Zone 4

I did the similar thing at Heathrow Airport. I took the underground from London to Heathrow, but I only had Zones 1 & 2 travel card, while Heathrow is in Zone 4. When I went out the train, I've put my worried face and I walked hurriedly to the guy on the exit gates (there's always a guy manually opening the door). When I got fairly near, I flashed my travel card asking in a worried voice "What's the quickest way to Terminal A?" Worked like a charm.