Yes you can unless restricted by owner (should be posted). You're Lucky, I can't carry a gun anywhere in my in my State...because I live in the "People's Republic of New Jersey" and I can't wait to move!!! (I do have a Fl non resident CCW)
You are correct that banks aren't off-limits in TN, (unless they are posted) however neither are local or state government buildings either, unless they are posted.
I'm sure you meant you had one but make sure you first have your Handgun Carry Permit (or other state equivalent). ON YOU. Carrying a loaded weapon in any manner is illegal in TN, the Permit is a legal defense.