Do it! There are a lot of opportunities in the NT for first job pilots. The basic list is as follows.
Territory Air Services
Chart Air
Air Frontier
Katherine Aviation
Kakadu Air Services
Air Arnhem
Gunbalanya Air Charters
Black Diamond Aviation
Marthakal
Arafura Aviation
Jandakot Flight Center
Northern Territory Air Services
Hardy's (Not a first job but advisable 2nd)
and any flight school if you have an instructor rating
Savannah Aviation in Burketown QLD is also a solid option for a first job on a Twin. GSL, Air Fraser and Daintree are in QLD as well.
Plenty more out there and a quick stalk of FR24 and 16right will give you a huge list of operators to target.
If you can't find something in the current climate you're doing something wrong.