All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His Slave and Messenger.
If someone wipes over his socks after invalidating Wudhu' (ablution), and then removes those socks after wiping over them, then he should not wipe over them if he puts them back on unless he washed his feet and then put his socks back on while in a state of purity resulting from washing (by performing complete Wudhu'), not from wiping. We say that this is correct, as you mentioned, because he did not wear his socks after having washed his feet. However, if he washes his feet and wears his socks and then removes them while he is still in the state of purity and puts them back on before his Wudhu' is invalidated, then it is permissible for him to wipe over them because he wore them in a state of purity resulting from washing. Wiping over the socks, leather socks or the like is not permissible unless one has worn them after making Wudhu' in which he had washed his feet with water.
According to a Fatwa of the Permanent Committee: “If you removed your socks after having already wiped over them in order to change into other socks, then it is not valid for you to wipe over the new socks if you wiped over your previous socks in your last Wudhu'. Your Wudhu' is also thereby invalid. You have to perform complete Wudhu' again by washing all the parts of the body that must be washed in Wudhu', including the feet, and then you may wear the new socks and start counting a new period of wiping…”
Shaykh Ibn ‘Uthaymeen said; “If someone removes his socks after having wiped over them, then it is not valid to wipe over them again. This means that he should not wear and wipe over them again unless he performs a complete Wudhu' in which he washes his feet…”
Allaah Knows best.