Conditions are extremely poor right now. Worst I've ever seen. It doesn't look like any significant snow will fall until your last day. I should be there on the 4th (unless it doesn't snow), but that doesn't help you much. You'll need to get up high. Without a guide, pretty much your only bet is to go to Blind Bull. It's a 30 mile sled ride from the trailhead, but you might find a little snow up there. Otherwise, your best bet might be to head up to Togwotee/Togwotee Pass. It's a good hour and a half windshield time, but closer to 10k feet in elevation.
There are many other high areas near Alpine, but you'd need a guide to get to them. I'm pretty sure there are guides available for a fee in the area. Maybe call Rockin' M Ranch? I know they do guiding.
I do agree somewhat to this post, and I do appreciate that you have referred me. It has been a warmer than normal winter. The fact of the matter is, it is a strange winter, but not necessarily a poor winter. There is still 110% of average snowpack over 8000 feet. I have been quizzing my renters daily, "how was your day"? Not a single person in the last 20 days said "the snow sucked", more often than not, I am hearing "It was much better than I thought it would be". If you are a regular here and only know the Greys River, you may have to get creative. Most locals, and myself ride everywhere but, the Greys, there are many other places to get higher faster. If one is willing to ride to Blind Bull, all reviews have been positive. About 14 days ago blind bull received 33" (per the groomers report) and 3 days ago another 8" on top of 1.5 feet of soft snow. The warm temperatures have fooled many into believing the winter is over. It is frustrating for me to explain this elevation game that we are in this season, many don't understand it. The bottom line is, riding is poor, plenty of snow but hard, under 8000 feet, but still surprisingly good over 8000. Togwotee starts at a higher elevation, but will be much more tracked up. Those who have ridden Togwotee and then learned the Alpine area know the difference. I am not putting Togwotee down, but for those willing to explore, there is no comparison to what Alpine has to offer. Many more awesome drainages in addition to the Greys River which are over 8000 feet---Sheep Creek, Indian Creek, Commissary, Willow, Strawberry, Dog Creek, North Murphy, and many more. Reports compared to the rest of the western US, our riding is pretty decent.