To add a new Hospital you need to either have the correct permission level, such as Administrator, or you need to contact the appropriate person. If you search the Resident Hospital list and the correct Hospital can not be found please go to the main navigation bar and select 'Hospital ' under Administration.

Here you add the Hospital for your facility. You only need to add the new Hospital once and it will come up in future in all locations searches.
You can now return to Resident Hospital search be able to find the Hospital and add it to your residents list.