Predicting missing links in incomplete complex networks efficiently and accurately is still a challenging problem. The recently proposed CAR (Cannistrai-Alanis-Ravai) index shows the power of local link/triangle information in improving link-prediction accuracy. With the information of level-2 links, which are links between common-neighbors, most classical similarity indices can be improved. Nevertheless, calculating the number of level-2 links makes CAR index not efficient enough. Inspired by the idea of employing local link/triangle information, we propose a new similarity index with more local structure information. In our method, local link/triangle structure information can be conveyed by clustering coefficient of common neighbors directly. The reason why clustering coefficient has good effectiveness in estimating the contribution of a common-neighbor is because that it employs links existing between neighbors of the common-neighbor and these links have the same structural position with the candidate link to this common-neighbor. Ten real-world networks drawn from five various fields are used to test the performance of our method against to classical similarity indices and recently proposed CAR index. Two estimators: precision and AUP, are used to evaluate the accuracy of link prediction algorithms. Generally speaking, our new index only performs competitively with CAR, but it is a good complement to CAR for networks with not very high LCP-corr, which is a measure to estimate the correlation between number of common-neighbors and number of links between common-neighbors. Besides, the proposed index is also more efficient than CAR index.