The clustering cofficient C of a network, Which is a measure of direct conuectivity between neighbors of the various nodes, ranges from O (for no connectivity) to 1 (for full connectivity). We define ex-tended clustering coefficients C (h) of a small-world network based on nodes that are at distance h from a source node, thus generalizing distance-1 neighborhoods employyed in computing the ordinary cluster-ing coefficient C = C(1). Based on known results about the distance distribution Pb(h) in a network, that is, the probability that a randomly chosen pair of vertices have distance h, we derive and experimentally validate the law Ps(h)C(h)< clogN/N, where c is a small constant that seldom exceeds 1. This result is significant because it shows that the product Ps(h)C(h) is upper-bouded by a value that is considerably smaller than the product of maximum values for Pb(h) and C(h).