Internet-of-Things (IoT) is featured with low-power communications among a massive number of ubiquitously-deployed and energy-constrained electronics, e.g., sensors and actuators. To cope with the demand, wireless-powered cooperative relaying emerges as a promising communication paradigm to extend data transmission coverage and solve energy scarcity for the IoT devices. In this paper, we propose a novel hybrid relaying strategy by combining wireless-powered communication and ambient backscattering functions to improve applicability and performance of data transfer. In particular, the hybrid relay can harvest energy from radio frequency (RF) signals and use the energy for active transmission. Alternatively, the hybrid relay can choose to perform ambient backscattering of incident RF signals for passive transmission. To efficiently utilize the ambient RF resource, we design mode selection protocols to coordinate between the active and passive relaying in circumstances with and without instantaneous channel gain. With different mode selection protocols, we characterize the success probability and ergodic capacity of a dual-hop relaying system with the hybrid relay in the field of randomly located ambient transmitters. The analytical and the numerical results demonstrate the effectiveness of the mode selection protocols in adapting the hybrid relaying into the network environment and reveal the impacts of system parameters on the performance gain of the hybrid relaying. As applications of our analytical framework which is computationally tractable, we formulate optimization problems based on the derived expressions to optimize the system parameters with different objectives. The optimal solutions exhibit a tradeoff between the maximum energy efficiency and target success probability.