Man, good luck…the “thin” part is what makes it challenging. Well, that and less than $15. 
If you need Small, then Pearl Izumi is blowing out their white Elite Tall socks for $5 / pair…they have other colors in other sizes, though.
I have been able to find thin, crew length single color socks occasionally at places like DSW, but never looked for white, so don’t know if that is a possibility.