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It should take approximately 25 minutes to hike a mile with decent wool socks, more or less time depending on the terrain and size of pack.
It will take 35-40 minutes wearing cotton socks — you need to add on time for fixing blisters and for all the stops you'll want to make because your feet hurt!
Socks for hiking need to be made of fibers that are resilient enough to cushion the feet and are elastic enough to fit closely without wrinkling and causing blisters.
Cotton has none of these desirable characteristics. If it is mixed with other fibers to try to make it work better, you still lose out in other ways. Good quality wool processed by modern methods has the desired resilience and elasticity.