Must one wash his hands after touching his socks?
No.[1] However, there are those who are stringent in this matter.[2] This especially applies if the socks are filled with sweat which have gotten onto one’s hands.[3]
[1] Kaf Hachaim 4:72 [brings opinions who state that one should wash his hands but concludes that even in their opinion it is merely an act of piety to be stringent to wash his hands after touching his socks as from the letter of the law washing is not required]; Ashel Avraham Butchach Hilchos Hanhagas Adam Baboker that touching sweat is only a problem than touching the actual skin however when touching a garment then it is only an act of piety to wash hands if sweat gets on them; Aruch Hashulchan 4:21“one who removes his socks does not need to wash his hands as it has the same status as the rest of one’s garments”
[2] Maaseh Rokeiach 2 Likkutim 1 in name of Rabbanim; Kaf Hachaim ibid
[3] See Sheivet Halevy 8:2; However, see Kaf Hachaim 4:98 in name of Ruach Chaim that one is not required to wash his hands after touching a sweat filled shirt, even if the sweat came from covered areas of the body. Ashel Avraham Butchach 4 rules likewise that one is not required to wash the hands, although he concludes it is an attribute of piety to do so. Halichos Shlomo 20:17 in name of Rav SZ”A is lenient and so rules Piskeiy Teshuvos 4:30; See Halacha F Q&A