The Items found in chests within the dungeons don't seem to be getting any better, if I'm in the Hall of Trials or the Jade Way, is this treasure offered based on something specific or have I just not gotten far enough into the game to get better things from the chest?
The items that you can find in treasure chests are fixed, for any given set of floors in a certain dungeon. So, yeah, you're going to need to unlock another dungeon to get better items from adventuring.
If you've gone awhile and haven't unlocked any new dungeons; make sure you aren't ignoring the events in the town square, pub and chapel. They may seem to have no effect, but a lot of the progression in the game depends on triggering certain events.