Many deformations occur in old RC structures, and the method of reinforcing them using PC members standardized and manufactured in the same mold has difficulty in coping with all deformations (construction error); as a solution to this, overlapping anchor shear connections consisting of three concrete segments have been proposed. This study proposes a special connection for strengthening existing reinforced concrete (RC) framed structures that are more than 30 years old with precast concrete (PC) members. The proposed overlapped anchor connection consists of two anchor groups installed in three concrete segments, which can accommodate deformation by adjusting the height of the concrete connection segment. In this study, the unique shear behavior of the overlapped anchor connection was investigated by analyzing the shear resistance of the single-anchor connection with one post-installed anchor and one cast-in anchor. Eleven specimens of anchored connection models were constructed and push-out tested, and the experimental parameters were analyzed. The shear strength of the single-anchor connection was calculated using a proposed design formula based on the American anchor design code, and the shear strength ratios of the experimental shear strengths to the calculated values were 0.93 on average. The proposed design formula can predict the shear behavior and expected effects of the proposed connection even in differing designs.