The Shrine Mountain Inn on Vail Pass is an easy two miles in. From there you have pretty easy access to the black lakes ridge, which is a great cliff/BC booter zone, although it definitely needs more snow at this point.
I've also heard that the Fowler-Hilliard hut has great skiing right out the door, but I've never been there.
Personally I would suggest against the huts closer to Leadville because I don't think that area generally gets as much snow.
As you probably know, the biggest problem you are going to have with the 10th mountain huts is that they are super popular and in general weekends book out months in advance.
I would also recommend checking some of the other huts and yurts which are not part of the 10th mtn div system located farther south in Colorado. For two reasons: 1) they don't book out as many months in advance. 2) There is actually snow in Southern Colorado at the moment.
Hope that helps!