Instead of descending from Grießenkareck by the ascent there is a fine highline-trail along the wide crest south to Saukarkopf (2014 mtr). This way provides a fine walk with a great view towards the Radstädter Tauern and the main ridge of the Alps.
From the top of Grießenkareck the good perceptible way goes south along the crest. It leads round an unnamed knob (2028 mtr) and then to Saukarkopf.
From Saukarkopf the way narrows and leads via Ahornkarkopf (1923 mtr) to a saddle about a kilometre before Mooskopf (1961 mtr). From there a good path leads south-west to Schüttalm.
A few minutes before Schüttalm you can take a path to the right, going back to Grießenkareck along the Bettlersteig (Beggars Path), a rather small way which provides excellent views along the Kleinarltal. Under wet conditions I would dissuade from walking this path because it's very slippery.
Not so exciting as the Bettlersteig but more comfortable you can descend to Schüttalm (where a hut is situated too) and then south-west to Kleinarl (way #20) and along the road back to Wagrain. The finer way instead leads from south of Schüttalm a few kilometres north-west (way #33) to a crossing. There you take the right way (#21) and a few minutes later again the right way (#21) until you can see Nazbauer again. At this point (it's called Obervierteck) a very steep and slippery (and adventurous ;-) path leads to the left. After about 5 minutes it will hit the street to Nazbauer. You go downhill for the first two serpentines, then the small path branches off to the right and will lead direttissima to Wagrain.
All ways and paths are very well marked and the signposts are excellent. The walking time from Grießenkareck to Wagrain will be about 2 or 3 hours.