If the cage will fit into the aviary then that should be great (again after quarantine of course ). Since adding on will effectively be changing the territory and making it new it shouldn't cause as many if any of the territory issues sometimes shown. If they are all showing interest or leaving each other alone (but no real aggression) then yes you could let them out of the cage into the aviary on the newer extension part (I realize it will all still be open to each other but will be a new "section". I would still keep an eye on them for another 24 or so hours to make sure there is no real aggression once there is no cage to keep them separated.

IMHO those changes "should" help make everyone settle in but sometimes you will have one or more that just won't accept each other so make sure you do have a plan for that just in case

Good luck!