I am a devoted fan of merino wool socks. That is all that I wear. I have a selection of these from Smartwool Adrenaline Light Mini Crew to Wigwam Wool/Silk Hiker crew.
The Adrenaline Lights are for splashing though streams with light, mesh trail runners, not worrying about getting wet as the thin, low cut socks and Montrail Hardrock shoes dry eventually and there is no concern in the warmth of summer.
The Wigwam Wool/Silk Hikers are my winter sock that make my feet feel like they never got out of the sleeping bag. They are full crew height and sooooo.. warm, soft and cushy! They require sizing up your footwear. I keep them dry in winter with Montrail Mid-Comp gore-tex shoes and wp/b gaiters.
I realize, in your post, you specifically mentioned "socks to go with trail runners." I am considering a new winter system with the same Wigwam socks but with Rocky Gore-tex socks over them and a pair of the Hardrock trail shoes, sized up for the extra bulk. I'm just having trouble getting my mind around wearing trail runners with snowshoes and crampons but, maybe I just need to give it a try.
Between these "extremes", I have Smartwool Adrenaline Crew socks. They are slightly thicker than the Adrenaline Lights and are full crew height for a little exta wamth in shoulder seasons. They are my go-to choice, particularly if most of my hiking will be along ridge tops where stream crossings are not an issue. This sounds like the conditions that you will experience on the AT in October.
Of course I always carry a pair of spare socks, safely tucked away in a dry spot in my pack for a little lightweight insurance!