Wireless is the best answer.

Problem 1: how to communicate with the public.

This has to be through something they already have and that is radio. It would be to great advantage for radio stations to be hardened for emergencies like this. If they are the only ones on the air, they have a huge devoted audience and can charge premium dollars to all business that want to advertise their emergency supplies to the public. The week link in radio, is the antenna. A temporary antenna could be made with a long wire and a helium balloon or bounced off a reflector on a hill side to suffice until the main antenna is repaired

Problem 2: hospital communications.

I left out emergency and law enforcement since they all ready have radio. Hospitals if not already setup for radio should be. Data on wireless has been around for decades. Better yet is wireless into the internet which gives hospitals access to all sorts of patient data, insurance, and colleagues around the world. This could also be in the form of wireless to satellites to internet. One last note, Sattleite cell phones would not be affected once a hurricane had past.