The next level i.e., rate of change of Jerk is called Jounce. However, I'm afraid I don't know how that would be described in car passenger's terms in your example.

I personally prefer snap over jounce, as that gives the progression snap, crackle, and pop.

See https://en.wikipedia.org/wiki/Fourth,_fifth,_and_sixth_deriv...

Cars have "jounce dampers", also known as bump stops.