Launceston and vicinity

Congratulations - you are about to do one of Australia's great walks; right up there with the best internationally. If you are doing the guided walk, look forward to the food and wine as well. Our guides were great company and dab hands in the kitchen as well.
As to your Launceston free day, I'd recommend a trip to the Cataract Gorge; lovely scenery just a short walk from the city (either the bush track on the southern side of the river, or the paved path on the northern side). Good spots for lunch when you get there, and it's quite a stunning site so close to the centre of a city.
Alternatively, if you're a fan of good wine like I am, there are some great wine tours in the Tamar Valley - Joseph Chromy at Relbia (about 20 minutes from town) is excellent, or you can do a bus tour along both sides of the river if you have the time. Try some of the excellent sparkling wines, Tasmania's outstanding pinot noir, and if at Chromy's they do a fantastic sticky.
Enjoy!!!

You could add the Sea Horse Aquarium at Beauty Point if you travel up the Tamar- a bit different!!
A bit further afield is the incredible drive up Jacob's Ladder at Ben Lomond National Park.
The drive up to Stanley and The Nut is also interesting.
A leisurely drive up through Mole Creek is also a possibility - great country and nice pubs etc.
Very envious of your planned trek.