In this paper, we consider $S$-chains of extremal self-dual and self-orthogonal codes and their applications in the construction of quantum codes. Then, by virtue of covering radius, we determine necessary conditions for linear codes to have subcodes with large dual distances and design a new $S$-chain search method. As computational results, 18 $S$-chains with large distances are obtained, and many good quantum codes can be derived from those $S$-chains by Steane construction, some of which improve the previous results.