How long as you going to be here?
Daywalks reasonably close: Roland (Roland to Claude can be done as a long day walk with a car shuffle, or a gentle overnighter from Gowrie Park with a bit of back-tracking); Dial Range; St Valentines; Badger Skyline; Minnow Falls; Walls (but check the weather first); Lees Paddocks; Long Tarns Track to Rinadena Falls; Meander Falls and the network of tracks from that area; Narawntapu; Rocky Cape; Leven Canyon but be careful if it's wet. The northern Tarkine coast isn't too far away and there's some great walking there with a mix of 4WD tracks, beaches and rock-hopping, plus some scattered short walking tracks such as Lake Chisholm, Trowutta Arch and Dip Falls. Lower level stuff from Pencil Pine (Cradle NP) including Hounslow Heath. Augusta Dam Rd is gated at the dam this time of year, but you can usually walk in from there and it's well worth the effort (well, I like it). Warners track to Adams Peak. Winterbrook Falls (and you can continue on a sidetrack up to the Black Bluff area if conditions suit).
Further east you have the NE Abels: Barrow, Victoria, Albert, Ben Nevis, Saddleback, Arthur, Ragged Jack and the Ben Lomond trio. Beware of ice. Good coastal walking at Mt William, and there's a good track up Mt Cameron.
Overnight: Depends on the conditions. Snow generally doesn't linger long except in the higher reaches, but ice is more common and more treacherous. Get a set of microspikes and you can get a whole lot further (they would help on the higher daywalks, too). There are heaps of tracks going up onto the CP from the north: Warners, Dixons, Mother Cummings, Syds, Higgs, Parsons, Yeates. Go up, find a good spot to pitch, and wander. You can also drive up to Lake Mackenzie and head south on various walking tracks, but be prepared for wet feet, snow or not. Most people do the beach traverse at Narawntapu as a two-dayer, but that needs a car shuffle. Lots of options in the Walls and snow shouldn't bother you much if you keep off the higher ground. Again, car shuffle required, but parts of the Penguin-Cradle Trail would be within your reach. Arm River track to near Wurragurra Creek, then up the valley and explore February Plains. You can also use the Arm River track to get into New Pelion (or a little further to Old Pelion) - how far afield you go from there depends on the conditions, but the track itself is good fun. Freycinet isn't a massive drive for people used to mainland distances, and the circuit there, including the sidetrips to Bryans Beach and the Mt Freycinet summit, is a great two-dayer. There are also some shorter tracks, and a lovely beach walk at Friendly Beaches.
There's a lot more, but that's a start. ListMap (
https://maps.thelist.tas.gov.au/listmap/app/list/map) is a great planning tool.
Just note that campfires aren't allowed in the WHA or NPs.