alta off-pisteThere are only a few places in Alta where it sounds as if you could get into trouble, i.e., steep with no nearby alternate way down...
Trying to ski directly under the Supreme lift and the Supreme Bowl, just to skiers right of the lift, is fun, but there are cliff zones and narrow steeps. You may want to think twice about going there.
High Rustler and the north rustler face (Eagles Nest, Garbage Chute, etc.) are locals' favorites, but you may want to pass on them initially. These are all at the very end of the high traverse off Collins. Again, there are cliff zones and narrow steeps, and consequently, some great powder to be found days after a storm. However, you may find this are a bit much.
High Yellow Trail and East Greely, on the east side of the spine to skiers right of the Collins lift are also challenging.
As others have suggested, Devils Castle is well worth the hike, immediately skiers right from the top of the Sugarloaf lift. The Apron is more typically reached by bearing skiers left from the Supreme lift. The trail into the base of the Apron has a bit of exposure, and you hike for most of your turns, but this is a spot that keeps snow well into the summer because of its sheltered exposure. The top of the Apron is fairly steep, and certainly seems that way as you climb the bootpack, but the snow is deep and soft.
Catherines, a hike to skiers right off the Supreme lift, offers a number of pitches of varying difficulty. The runs are shortish but well worth it. I like going far out, because there is less skier traffic.
Now that there is a HSQ to the top from the Wildcat base, the runs off the old Wildcat double are skied far less than they used to be, and the Wildcat woods, to skiers left of the lift, are full of fun pitches and powder stashes.
One thing about Alta is that the width of skiable terrain served by most of the lifts is often wider than the lift is long, even when you avoid overlapping terrain served by multiple lifts. Even when it looks as if the local powder hounds have harvested everything, there are lots of stashes left in the cracks between the main runs.
Enjoy your trip...