>>"The oxygen hose from the helmet hooks up to a hose in the jet."<<

Technically, the pilot's oxygen connection is attached to the ejection seat and the ejection seat get its oxygen from the jet.  In the event of an ejection, the seat is equipped with a small oxygen bottle good for 5 to 10 minutes and it automatically switches to the internal bottle when you pull the ejection handle.  The 5 to 10 minutes is long enough to survive the ejection at high altitude till you drop down to a low enough altitude to breath on your own.  If you land in the water, you can actually breath underwater for a short time till you come to the surface.  That's why the post ejection procedures over water are different than overland.  Overland you want to loosen you oxygen mask before ground impact in case you're knocked unconsious.  If you're unconcious when your seat oxygen goes empty, your dead.  But when you eject over water you cinch up your oxygen mask straps to make sure it has a water tight seal around your face.  You can breath underwater till you come to the surface.  Once at the surface, you loosen your oxgen mask or dump it entirely.