If you are driving keep the cage in the middle of the back seat.
When placing it in the back seat make sure the cage is in a box or holder that will protect the seat covers on the seat and back of the seat ... they love to nibble on anything they can reach! Or at least a board between the seat and side of cage.
The reason for keeping it centered in the back seat is that it will most likely be out of the direct sunlight while driving/parked. Sunbeams + glass window = intense heat, even in the winter. When using an open carrier (wires on all 4 sides) we place it in a box that can be securely belted in to keep it from moving around should we have to stop suddenly, but with the top open for fresh air.
Do not place in the floor near an air vent! Chinchillas prefer the cool rather than heat. They can easily over heat and die.
Do not get a cloth carrier (like the ones made for cats or small dogs), they will chew through it in no time!
Jo Ann