Unlocking hospital actually related in a way with finding Sophia so I'll explain the latter first.

 After you assist Sofia in arresting Mario, she'll invite you to her house. But after 2 or 3 visits, suddenly she dissapears and you'll find a memo on her desk. 

However, make sure to have either Athlete or Living Weapon Trait activated first before the next step, which is going to Evergreen and you'll find a new pathway called Shack. There you'll find 2 of Mario's goons protecting the Shack and they want to beat you up. If you already have Athlete or Living Weapon Trait, you can beat them uo instead and inside of the Shack you'll find Mario keeping Sofia as hostage. You can rescue Sofia by beating Mario up or kill him. After that, you'll take Sofia to her home and you'll get new scenes with her. 

As for unlocking the Hospital, basically you have to lost a battle to unlock that. If you don't have any of the 2 said Traits before challenging Mario's goons, you'll get beaten up and sent to the Hospital automatically. The same way also applies when you lost a battle during the Spirit Slaying Minigame. 

If you're wondering about the new scenes in Hospital, there are 2 new characters that you can get the scenes from. First is the Nurse ( you have to be sent to the hospital after losing a battle 3 times ) and the Doctor ( just buy 2 of her experiments and ask her to help you test the after-effect )