Can an aircraft re-pressurize at high altitude?
Pressurization is controlled by two processes: air in and air out.  Air is constantly being let out of the cabin through (at least) 2 outflow valves, typically located at the aft limit of the pressure vessel.  Air is let into the cabin through the air cycle machines (a.k.a the "packs") which are provided with very hot and high pressure air from the compressor stage of the engines and eventually output into the cabin through the air conditioning vents.

During normal cruise flight, the mass flow rate of air into the cabin is constant (because the flow rate through the engine compressor is constant), thus pressurization is maintained by actuating the outflow valves.  Open them up a little more to raise the cabin altitude, close them up a little more to descend the cabin altitude.  Note that while maintaining a constant cabin altitude the outflow vales are partially open -- the incoming air is more than adequate to pressurize the airplane.

When a main cabin door or something bigger is opened, the airplane will no longer maintain pressurization and everyone must don oxygen masks to avoid unconsciousness.  With a planned cabin dump, masks will be donned before the cabin is evacuated, which will happen before any doors are opened.  As noted by others, the pilot oxygen is fed from cylinders of pure oxygen and will last a long time.  Passenger emergency oxygen is from chemical oxygen generators and lasts 12-15 minutes.  I'm not sure what type of oxygen masks the HALO jumpers would wear, but presumably they have portable oxygen for the jump not related to the airplane oxygen facilities.

When all of the jumpers are away the doors are closed and once again the airplane is capable of pressurization.  The outflow valves will be commanded closed and the cabin altitude will start descending.  Once the cabin altitude is back down to normal levels, the crew will remove and stow their oxygen masks.

There is no need to descend to accomplish this because the air provided by the engines is much higher pressure than even sea level air.  Re-pressurization is accomplished by closing up the airplane and letting air accumulate in the cabin, thus raising the pressure.  Once the target pressure is realized (~8000 ft), then the outflow valves partially open to stop descending the cabin and to maintain the target pressure.