Due to a common glitch, it is possible to exit the flying car by running to the left or right. It is very easy to do this glitch. This may be easier to accomplish by using consumables. This can result in mice's death, or mice escaping the cart in order to get the cheese and jump to the hole without help from the shaman.
The mice can push the car to the cheese, and after the cage begins to float down, push it to the hole. You can also push the car to a side of the map until the side plank disappears and jump into the hole.
The safest technique for the shaman is to summon one ghost (spaced) anvil before you reach the cheese. This way the car will rise slowly, making it easier to summon more anvils (3 ghost anvils should push the car down) once you reach the cheese. If it looks like you are going to fall too fast and aren't lined up to hit the mouse hole, a spirit can push you where you want to get. It is advised not to use a drag ball as the car will usually go too far, often off-screen where you can't get back. You can also summon a large Global anchor plank under the car to give you time to build something.