The Kötschach valley offers wonderful walking opportunities into the Nationalpark Hohe Tauern throughout the year. Drive through Bad Gastein and follow directions to the Hoteldorf Grüner Baum, where you'll find plentiful car parking. You can also take the bus from Bad Hofgastein directly to the hotel (don't forget your discount with the Gastein card).
Lots of horse and carriages will pass you in summer and especially so in winter. You can hire one from Fam. Fiaker Lederer, the journey to the end of the valley takes around 1 hour, you can eat at the Alpenhaus Prossau, before your return journey (total time 3 hours). In winter you should check for avalanche warnings before undertaking this hike.
Follow the Luis-Trenker-Weg which leads through the romantic Malerwinkel to the Cafè Himmelwand. The path continues through woods, alongside the Kötschachter Ache stream.