Did the complete TAT in 2011, had three pairs of socks, one smart wool and two road gear socks. Tried to wash out the pair I rode in every night. Never wanted for more socks, did end up wearing the smart wool ones more as my boots weren't completely broken in and after one day of long walks the heavy socks helped, plus lots of band aids and mole skin.
That reminds me bring some band aids and mole skin to help with any foot issues!!!!
check my TAT link for packing list, served me very well for a LONG extended trip. I carry less on short rides less.