As the sun set and the shadows grew to their longest, they broke up camp and packed their stuff on Appa. It was now the evening of the first day after the storm. They didn't know that they hadn't been even declared missing by now. Would they have waited some more, they would have been detected by the search-party, Azula encountered a day later. Luckily Toph had made sure that their camp would not be recognized as such. So the search-party wouldn't even find traces of their survival. They refilled their water bottles to make sure they had enough water to keep up their disguise, once they reached the main-land.
Again they had to split up and fly in turns. This time Aang and Katara would stay on Appa and fly the rounds. It took them half the night until all of them were back together. They had landed on a bigger island in the east, closer to the main-land. From there on they could move from bigger island to bigger island until they would reach their destination. The island only contained small villages and a rough surface with a lot of cliffs and waterfalls. The people living there weren't as informed about the war and the propaganda common on the main-land, so they were quite sure to pass unrecognized. Only Aang's arrow could turn out to be a problem.
Aang didn't want to listen to reason, though. He didn't want to hide his heritage, despite they told him it would endanger their mission. Iroh and Hakoda both had to talk to him over an hour, until he finally agreed to wear a hat to cover his bald skull. All of them used the clothes they had from the ship, to disguise themselves as Firenation-citizens and got rid of everything that would reveal their true identity. It wasn't easy for most of them. But since they had not much to survive on their own, they needed to enter a village to find shelter and food.