That's not the same as "all subsequent deaths in that hospital" though, is it?
If you die in hospital after a serious road accident, that will not be recorded as a coronavirus death, even if others have died from the virus in the same hospital.
The end of June is a decisive moment, as that is when players contracts are up. Nobody, certainly not the EFL, can interfere with the individual contracts between Clubs and players unless by agreement with all concerned. The big push will be to meet this deadline, somehow.