I've done the entire track... in a tent.
However, I reckon you could do it, as I remember there are plenty of trees about at all of the marked campsites. I recently did a week in the Tarkine rainforest (NW Tas), and my biggest problem was finding enough open ground between trees to pitch the hammock and tarp but I always found a few spots that suited.
Keep in mind that there are huts as well, so you have alternatives. Might be worth keeping a lw pad at the bottom of your pack in case you need to spend the night in a hut or on the ground.