I'll add my vote to the previously mentioned Pine Valley + South Cape Rivulet.
Just spent the June Long weekend up at Shadow Lake (well there was less rain up there than Hobart!), which was really pretty, even in the mist + drizzle.
Mt Field area is worth exploring (there are plenty of seldom-populated areas once you get past the Tarn-Shelf day-walkers).
Arm River Track is a goodun - and Pelion hut is a good base camp with numerous options depending on the weather.
The Walls is pretty magical too - recommend a circuit into Dixons Kingdom via Lk Adelaide / Lk Ball, then out the main track.
I suppose all these highland options carry with them a disclaimer in terms of snow-cover and being aware and sensible of any deteriorating weather conditions.
