Yosef Karo, Translation:Shulchan Aruch — Orach Chaim 636…
“ An old sukkah — that is, one that was made more than thirty days before the holiday — is valid provided something new was done to it for the sake of the festival. For this, even a handbreadth by a handbreadth [of schach] is sufficient if it is in one place. If the new thing goes across the whole sukkah, even a little bit is sufficient. ”
