In a multihop Wireless Sensor Network (WSN), a salient point among routing protocols that do not depend on network topology and existence of neighboring nodes is the need to know sensor node’s geographical location with respect to the sink node. This is obtained by some means like Global Positioning System (GPS) and localization techniques. In a prior work, we have proposed RSSI-based Forwarding (RBF) protocol that works without knowledge of node’s location by using a Received Signal Strength Indicator (RSSI) level of beacon signals transmitted by the sink. Through contention, a next-hop node is determined among the forwarding candidates using a timer-based suppression scheme. We propose an improvement of the suppression scheme in which a contender closer to the sink is favored with a higher probability for being selected as a next-hop node. By means of simulation, it is shown that the performance of RBF is significantly improved using the enhanced mechanism.