In two-tiered Wireless Sensor Networks (WSNs) relay node placement is one of the key factors impacting the network energy consumption and the system overhead. In this paper, a novel connectivity-aware approximation algorithm for relay node placement in WSNs is proposed to offer a major step forward in saving system overhead. Specifically, a unique Local Search Approximation Algorithm (LSAA) is introduced to solve the Relay Node Single Cover (RNSC) problem. In this proposed LSAA approach, the sensor nodes are allocated into groups and then a local Set Cover (SC) for each group is achieved by a local search algorithm.