Great question — answering as the UphillTech team:
Well, the good news is you'll either have to deal with black flies or the last of the snowpack, but you won't have to deal with them both at the same time. May is peak black fly season : N.H. Black Fly season If you are above 4K feet and there is a storm, significant snow is a possibility. As far as the old snow pack goes, it should be pretty well set up corn and snow shoes won't be needed. If you plan on early morning starts, some kind of traction aids and ski poles should cover whatever you run into. The Mt. Washington summit weather observatory is definitely where the worst weather occurs. Mt. Washington Weather Archives That link gives historical data that should give you some idea of...
Full answer by Fred the Magic Wonder Dog: https://outdoors.stackexchange.com/a/24894 (CC BY-SA 4.0).
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