There is a good trail leading up all the way to the base of mountain. The lower section of the black line shows still some trail, higher up you have to use your route finding skills and there are several options.
We scrambled up via black line, which put us about 100 feet below the main summit - somewhere on the ridge between North Snowmass and main Snowmass peak - good route if you want to do both peaks.
Red line - I have not actually not used, but have seen climbers descending in that gully.
Green line - we used for descent (it was easier on Duchess) - the top was not as steep, still a lot of loose rock.
I believe that the gully to the left of black line would be pretty easy too, did not explore that area too much.
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