In mobile ad hoc networks (MANETs), the network topology changes frequently and unpredictably due to the arbitrary mobility of nodes. This feature leads to frequent path failures and route reconstructions, which causes an increase in the routing control overhead. Thus, it is imperative to reduce the overhead of route discovery in the design of routing protocols of MANETs. In this paper, we propose an estimated distance (EstD)-based routing protocol (EDRP) to steer a route discovery in the general direction of a destination, which can restrict the propagation range of route request (RREQ) and reduce the routing overhead. In the EDRP, the change regularity of the received signal strength (RSS) is exploited to estimate the geometrical distance between a pair of nodes, which is called the estimated geometrical distance (EGD). Simulation experiments based on a random waypoint (RWP) model show that the EGD can effectively reflect the actual distance when it is less than the expected value of the distance [which is called the estimation radius (E-Radius)] between any node pairs. We also propose an estimated topological distance (ETD), which is a topology-based EstD, as an aid to the EGD, which can mitigate the effect of inaccurate EGD. The EstD is a combination of EGD and ETD. In the protocol, every node evaluates the link quality through the computational process of the EGD to eliminate the weak links and then uses the EstD (EGD and ETD) to steer the RREQ packets toward the general direction of the destination. Simulation results show that the proposed protocol can significantly reduce the routing overhead and improve the routing performance in dense or high-mobility networks.