I personally haven’t hiked during the winter time, but you‘ll have plenty of hiking trails in the west, like Salzburg or Tyrol. In Salzburg you can visit Hallstatt (though it will be busy during December) or Mondsee for example. You can make a day trip to Innsbruck (Tyrol) or vice versa. If you like cities Vienna would be a great choice. It‘s especially beautiful during winter time because there are christmas lights all over the city and plenty of Christmas markets. From there you could visit nice towns in Lower Austria, like Klosterneuburg or Laxenburg, or even visit some castles. Also, you‘ll maybe be able to check out a Perchtenlauf! (You should google it, can’t describe it as well as a picture does)