From the NWS Riverton, WY office — mentioned in their Area Forecast Discussion on Wednesday afternoon:
FRIDAY NIGHT INTO SATURDAY...AS THE ENERGETIC UPPER LOW MOST LIKELY FROM WESTERN WYOMING INTO NORTHEAST WYOMING...THERE SHOULD BE A LARGE SWATH OF PRECIPITATION BEGINNING TO WRAP AROUND THIS SYSTEM. I HAVE HIT THE NORTH HALF WITH LIKELY POPS FOR FRIDAY NIGHT AND EVEN A BIGGER AREA SATURDAY AS WRAPAROUND FURTHER IMPROVES THE CHANCES IN AREAS LIKE THE WIND RIVER BASIN. DETAILS ARE STILL UNCERTAIN BUT OVERALL BIG PICTURE IS SLOWLY COMING INTO FOCUS. HAVE STARTED SNOW LEVELS FAIRLY HIGH THURSDAY EVENING BUT LOWERED THEM TO AS LOW AS 8000 FEET IN YELLOWSTONE BY FRIDAY MORNING RISING TO AROUND 10000 FEET IN THE GREEN MOUNTAINS. HOWEVER...IT IS POSSIBLE IN THE SOUTHWEST THAT TEMPERATURES COULD COOL MORE THAN EXPECTED AND LOWER SNOW LEVELS IN THAT AREA LOWER THAN SOUNDINGS OR H7 TEMPERATURES WOULD INDICATE. ON FRIDAY...SNOW LEVELS WILL GENERALLY BE BETWEEN 9000 AND 10000 FEET.
THE POTENTIAL FOR SIGNIFICANT SNOW ACCUMULATION DOES EXIST FROM THIS SYSTEM ESPECIALLY ACROSS THE BIG HORN MOUNTAINS FRIDAY NIGHT AND SATURDAY. HOWEVER...EVEN OTHER AREAS IN THE WESTERN MOUNTAINS COULD SEE DECENT SNOWFALL...ESPECIALLY THE WIND RIVER MOUNTAINS AND ABSAROKA MOUNTAINS. STAY TUNED FOR FURTHER UPDATES ON THIS MAJOR CHANGE TO COLD AND UNSETTLED WEATHER.
My plans are to either depart around 1pm from Dodge City tomorrow afternoon and try to make it up to Yellowstone as fast, and safely as possible, taking a few 2 hour power naps along the way… -or- leave as originally planned very early friday morning and plan on my original stop in Dubois. I’m almost leaning more towards the Dubois scenario, because they might see as much snow as Mammoth given the elevation, and I love snow 
Then I’ll stay another night in Dubois Saturday night and hopefully by Sunday I can start my way to Yellowstone and make Sunday a nice little snowy landscape shoot most of the day. If the sun can come back out Sunday, it could be spectacular.