Praise be to Allaah.
First: A Muslim must be proud of his religion and not feel ashamed of its rites and rituals being performed in front of others, or of what shari’ah imposes upon him. If he fears that seeing the implementation of a religious practice or ruling may cause the unbelievers to have an aversion to the religion, then he wisely conceals, aiming at presenting Islam gradually and in an appropriate way. This doesn’t mean to be ashamed or weak or feeling disgrace from implementation of some religious rituals.
I suspect you fear from their disapproval because of their misunderstanding, inverted senses, and spoiled opinions, ideas, and inclinations. How could the removal of one’s socks and the washing of one’s feet be considered filthy or disgusting, when the fact that is that it effects cleanliness and the removal of sweat and unpleasant odor from the feet?
Second: It is not permissible for the person to pass wet hands over the socks unless he put them on in a state of purity. You can find the evidence in the following Hadith narrated by Al-Mughirah Ibn-Shu’bah, may Allaah be pleased with him, in which he said:
"I was traveling with our Prophet Muhammad (peace be upon him) and the Prophet asked me saying, "Have you got any water?" I said, "Yes" … and poured out water so that the Prophet washed his face and hands... and his arms then wiped his head. Then I bent over to pull off his khuff (a type of thick sock) but the Prophet said "Leave them for I put on my khuff in a state of purification" then he wiped over them. (narrated by Al-Bukhari, Fath ul-Baari No. # 5353)
Ibn Hajar, may Allaah bless him, quoted the following in his explanation of the portion of this Hadith in which the Prophet’s said "I put them on a state of purification":
Al-Hamidy related in his Musnad that the Prophet was asked whether one could pass wet hands over the khuff, the Prophet answered saying: "Yes if the person puts them on in a state of purification."
Ibn Khuzayma said that according to Hadith narrated by Safwan Ibn Assal, "the Prophet ordered us to wipe over khuff if we put them on in a state of purification, for 3 days during traveling and for one day and night in residency".
Safwan’s hadith, although it is sahih, does not meet the conditions defined by Al-Bukhari. But al-Mughirah’s hadith is in agreement with it in its indication of the stipulation of being in a pure state before putting on the socks.
Third: It is in fact easy to solve your problem—all you need to do is perform the complete ablution at home before going to work, washing your feet, then put on your socks. Then, if you want to perform ablution again at work, you can pass your wet fingertips over the back (i.e., top) of each foot.
And Allah Knows Best
Sheikh Muhammed Salih Al-Munajjid