Start without climbing skins, perhaps with a bit of nordic wax. From the front of the 10th Mountain Division Hut, ski S into the large clearing, then S across the clearing to intersect a stream in a shallow gulch. The trail splits at this point, with the left (E) fork following routes to trailheads. For your route, take the right (W) fork and follow the standard marked 10th Mountain trail as it traverses SW to a small pond, then drops through a clearing S to the North Fork West Tennessee Creek (11,100 feet), your second drainage crossing from the hut (count them as you go). Continue S as you climb a small ridge (11,130 feet) that separates the North Fork West Tennessee Creek from the west fork of West Tennessee Creek. Next, drop S through timber to 10,980 feet in the beautiful open valley of West Tennessee Creek. Navigation in West Tennessee Creek can be difficult when visibility isn't the best. Cross the middle fork of West Tennessee Creek, then climb 1/4 mile SW to a flat marshy area at West Tennessee Creek (11,080 feet).
Swing E for 1/4 mile through the marshy area, then enter a trail cut in dark timber. Follow this trail E, then turn right (S) and climb to a small lake. From this lake climb to a small saddle (11,140 feet), then drop S through a clearing into a marshy area in a subsidiary drainage. Cross S through the marshy area and continue descending SE to intersect the Colorado Trail at 10,880 feet. (At this point the Colorado Trail is marked on the USGS map as the Main Range Trail. To add to the confusion, it is also the Longs Gulch Trail in the USFS Tennessee Pass ski touring trail system.) Turn right and follow the trail as it leads SW for 1 1/4 miles up Longs Gulch to 10,900 feet. Here you'll see some possible avalanche slopes on the north side of the drainage. Avoid these by staying in the timber to the south of the trail. Regain the trail in the timber and follow it as it climbs SW to a flat saddle with several small lakes.
Drop S into Porcupine Gulch from the saddle and cross Porcupine Creek at 11,240 feet. Still counting drainages? Climb S out of Porcupine Gulch, pass timberline, and continue climbing to 11,800 feet on an alpine shoulder. This is the high point of your route. The climb out of Porcupine Gulch switchbacks up through a slice of timber to avoid avalanche slopes to the right, then stays high to pass above other avalanche prone slopes. Take extra care to choose a safe route here, as you could easily deviate into avalanche terrain. The trail as marked on the text map is accurate, while the trail on the USGS map is inaccurate. If you have time, take a scenic detour at the shoulder and climb point 12,313.
It's all downhill from the shoulder to Uncle Bud's Hut, so strip your skins. From your position on the shoulder, ski E down to timberline at a sparsely timbered saddle (11,680 feet). At this point it is very important to swing W and stay in Bud's Gulch (unnamed on the USGS map, marked on the text map), since it's all too easy to ski the tempting glades down into St. Kevin Creek and end up far from the hut. A compass check can help you here. Skiing down Bud's Gulch takes you through timber for 400 vertical feet to 11,360 feet where you broach an elongated low-angled clearing. The 10th Mountain marked route follows the Colorado Trail down the left side of the clearing to 11,260 feet. You then swing E and climb through sparse timber 1/8 mile and 120 vertical feet back up to the hut.
This last little climb can be bothersome after all the drainages you've trudged through. On the map it looks like you can contour to the hut from the upper end of Bud's Gulch, but this route is blocked by some fairly dense forest. The alternate that works—if you are good with navigation—is to ski the wide rib dividing St. Kevin Gulch from Bud's Gulch. Stay to the east side of the rib in light timber and traverse the east side of several bumps. Again, don't drop down into St. Kevin Gulch. The hut is only visible from several hundred feet away on either route.