You can hop on any non-reserved Ordinary class car you want and take a seat with a Green Pass.

On Hokkaido Shinkansen departing Shin Hakodate, had to book a later train because Green class was sold out on the departure I wanted , Sunday morning in April. Ordinary class wide open.

Weekday evening Morioka south bound, Green seats sold out. Ordinary class wide open.

Hokuriku Shinkansen Karuisawa to Tokyo, Green seats sold out all afternoon (2 - 7 pm) on a Sunday last November. All reserved seats sold out.

On Kyushu Shinkansen, Green class consists of 1/2 a car, so they sell out fast.

If you don't have a reservation in any class, you will be asked to move to the unreserved ordinary car (if one is available), if the conductor is doing his/her job properly.

Also Reserved seats in Ordinary Class are wider than unreserved Ordinary class seats on some trains.