Although Partial Relay Selection (PRS) for amplify-and-forward relay systems requires only the knowledge of Source-Relay (S-R) channels, in general, it incurs a significant performance loss. In cooperative systems with all single-antenna nodes, irrespective of the number of relays, the diversity order of PRS is limited to only one. This paper proposes a novel relay selection method for improving the performance of PRS scheme. In particular, as in the conventional PRS scheme, the relay that gives the best first-hop Signal-to-Noise Ratio (SNR) is selected. However, this selection is made from only a subset of relays, for which the corresponding S-R and Relay-Destination (R-D) links are not in outage.