When I watched the first season of Jojo no Kimyou na Bouken: Stardust Crusaders, I have never seen Joutarou fly with his stand, but suddenly in season 2 of Stardust Crusaders when Joutarou confronted Dio, he did fly. You could see on this scene, they did fly when they punch each other. Is this new ability of their stand?
They aren't flying, but rather a consecutive set of "Stand leaps" combined with The World's ability to stop time. This is basically when the stand uses its power to perform a very powerful jump by either jumping or punching walls and the floor. For example, in Jotaro's fight with Dio we see him punch a wall with Star Platinum to propel him in another direction.
The instance where The World and Star Platinum attack each other and both Stand users start to "fly", it is like lots of mini Stand leaps as they propel off each other. The 2 Stands punch each other so fast they begin to rise, because the forces acting on the other are equivalent to Stand leaps.