To chill down an engine, cryogenic propellant is pushed through the engine. This is fine for the first stage because the chill down propellant can just flow out the nozzle and on to the ground/the atmosphere. The upper stage, however, is closed up. How do you chill down the upper stage engine if there is nowhere for the propellant to go after it exits the nozzle?
Different vehicles use(d) different schemes.
The S-IV stage on the Saturn I vehicle had overboard vents for the LH2, and fabric blowout panels for the LOX. One of the LH2 vents is indicated, the blowout panels are the half-ovals at the lower end of the stage.
The Saturn V second stage used a recirculating chilldown system that didn't require overboard venting
A chilldown of the S-II stage engines begins prior to liftoff with the chilldown of the thrust chambers. Propellants are circulated though the pump and feed lines during first stage operation until a few seconds before first plane separation.
The Saturn V third stage also used a recirculating system.
The engine propellant pumps and gas generators must be chilled prior to start. This is accomplished during S-IC boost phase. LH2 is circulated, Figure 22-17, by means of stage mounted pumps through the engine LH2 feed lines, engine LH2 pumps, and gas generator LH2 bleed valves and then returned to the container. LOX is circulated, Figure 22-18, by means of thermal convection through the engine LOX feed lines, engine LOX pumps and gas generator LOX bleed valves and returned to the LOX container.
Source: NASA TECHNICAL MEMORANDUM X-881, APOLLO SYSTEMS DESCRIPTION VOLUME II SATURN LAUNCH VEHICLE
There is plenty of space to go for the propellant. For example, the nozzle of a vacuum engine itself is a giant empty cone. Then there's the interstage.
On rockets which perform hot staging, the interstage must necessarily be open.
And if all else fails, you can always just punch a hole into the interstage. I mean, obviously if someone builds a rocket, they will think of this. It's not like they perform the first orbital launch and only realize 3 minutes into flight that they never figured out how to chill the second stage engines.