Department/Team/Franchise Setup
It sounds like departments, teams, and sub-teams could be set up as contact sub-types of the organization contact type, possibly with custom relationships between each that are intuitively named, as described here:

https://civicrm.org/blogs/mjpforsberg/using-contact-subtypes-clutter-free-life

The hiccup with that setup might be trying to reliably get memberships to attribute from the parent organization through those relationships to the individual contacts themselves.