We propose the asynchronous receiver-initiated MAC protocol with the stair-like sleep mode; each node reduces its own sleep time by the sleep-change-rate depending on the number of hops from the source to the sink in wireless sensor networks (WSNs). Using the stair-like sleep approach, our protocol achieves high delivery ratio, low packet delay, and high energy efficiency due to the reduction in idle listening time. Our protocol can formulate the upper bound of the idle listening time because of the feature that the sleep time decreases in a geometric progression, and the reduction of the idle listening time is obtained by using the stair-like sleep approach. In our proposed scheme, the sink calculates the sleep change rate based on the number of hops from the source to the sink. By using the control packets which have the role of the acknowledgment (ACK), our proposed protocol can achieve the stair-like sleep with no additional control packets. In addition, even in the network condition that multi-targets are detected, and the number of hops to the sink are changed frequently, our proposed protocol can change the sleep change rate adaptively because the sink can always obtain the number of hops from the source to the sink. Simulation results show that the proposed protocol can improve the performance in terms of the packet delivery ratio, the packet delay, and the energy efficiency compared to the conventional receiver-initiated MAC (RI-MAC) protocol.