I'm not sure I understand.
Whenever you need to fly into a different city, you use the ATM-like machine that gives you a boarding pass. This ATM will let you pick a destination and time of flight—the time serves only one purpose, copyright protection. You have to match the time and destination with a code found in a table inside the "Leisure Suit Larrys Greatest Hits and Misses - Manual" PDF (not in the "Leisure Suit Larry 5 - Manual" PDF). It's on page 24.
Once you enter the correct password, you get a boarding pass. You enter the waiting room, show your AeroDork card to the security camera, wait a few moments for the announcement, and then use the boarding pass on the scanner next to the door, after which Larry automatically boards a plane and flies to the destination you chose.
If you somehow lost the boarding pass when you loaded a save game, you should be able to leave the airport, visit the ATM-like machine again, pick a new destination and time, re-enter the code, and get a new boarding pass. But I've never experienced losing it, even though I'm pretty sure I've loaded a save made in the airport at some point.
Unless there's some glitch I don't know about. Maybe save your game just before getting a boarding pass, and then after landing at your destination? You do have to solve some puzzles involving the airport, but that's only after you land in a new destination.
I don't think there's anything you can do at the airport when leaving an area, other than maybe reading some of the funny advertisements you might have missed, so I can't imagine it ever being absolutely necessary to make a save at the airport of a city you are departing from. If it's bugged, doing it in one go shouldn't take you more than two or three minutes.
