Tasman NP has lots of daywalks - Cape Raoul, Cape Hauy, Lime Bay, Stinking Bay, Mt Brown and Crescent Bay, Macgregor (a number of routes), Deep Glen Bay, Cape Surville, various bits of beach, etc etc. There's a track all the way from the Blowhole down to Fortescue Bay (longish two days to go down and back) and that links with the Cape Pillar circuit (part of 3 Capes that can be done free as an overnighter).
kunanyi-Mt Wellington has tracks all over the place and there are so many options up there, snow or not.
Mt Field is great any time of year and there are good daywalks as well as overnight options.
Bruny Island has some good daywalk options. East Cloudy is marvellous.
Understandable if you'd rather avoid the more exposed ranges (although it wouldn't be hard to borrow snowshoes while you're here

), but many of them have tracks that can be followed part way: Lake Skinner up in the Snowy Range, for instance, or the lakes at Hartz.
Even the SW has options in reasonable weather - Needles, Tim Shea, Sentinels, Twelvetrees, Eliza, Lake Mudd ...
Mt Dromedary and Platform Peak near New Norfolk.
Waterfalls all over the place - just google "waterfalls of Tasmania" and go for it.
Snow generally isn't a problem until later in winter and quite often we get extended periods of clear, settled weather in May and June; it's a great period for visiting some peaks.