Asana of course will answer your question but from my experience I would be surprised if another hierarchy is added. There has been some discussion of adding something equivalent to stacks like Evernote uses to organize projects. As you know the structure currently is:
Organization
Team
Project
Sections
Tasks
Subtasks
My immediate reaction is to consider prefixes and/or custom fields. Use an easily identified Prefix as the name of the Department or possible the Team within the Department. An example would be ACTY for Accounting Department and maybe there is an Asana Team called ACTY| Annual Audit. Your could decide to not use the Asana Team level for department but instead your Team with a prefix. You could also use Asana Teams as Departments like you do but put a Prefix on the name of Projects that identify the Department for better reference. You can also be creative with custom fields to identify your definition of Team. Just remember my link Team and Project Name Duplication Problems With Team and Project Conversation Email Address (Warning - #6 by James_Carl if you plan on using Team or Project Conversations. There are many suggestions on reporting. I have touched on a couple Managing/Prioritizing/Reporting A LOT of Projects in Asana - #2 by James_Carl Managing/Prioritizing/Reporting A LOT of Projects in Asana - #4 by James_Carl but certainly there are other ways.
Sounds like you also would compliment all with a number of Saved Favorites in Advanced Searches.
The book I referenced is very good. I do not know your business but do you have to review every Team and Project in detail or do you have trusted managers that can report your “Stucks” while you still being able to review as much detail as you desire individually. Seems like it would be a long meeting to go into so much detail. Many look at the available hierarchy outside of Organization as a number count of levels available and Custom Fields as an additional level. Hope some of this helps.