We investigate proactive eavesdropping with one half-duplex legitimate monitor (E) in unmanned aerial vehicle (UAV)-enabled suspicious relaying systems, which consist of a suspicious transmitter (ST), a suspicious UAV-based relay (SU) and a suspicious destination (SD). Under this setup, we propose two jamming-assisted eavesdropping strategies, namely, “Eavesdrop-Then-Jam” (ETJ) and “Jam-Then-Eavesdrop” (JTE), to maximize the eavesdropping throughput of E. Specifically, for ETJ, E first eavesdrops the suspicious signal of the ST in the ST-SU phase and then jams the SD in the SU-SD phase; for JTE, E first jams the SU in the ST-SU phase and then eavesdrops the suspicious signal of the SU in the SU-SD phase. However, in both strategies, as the jamming power of E (independent variable) changes, the SU can also adaptively adjust its deployment (dependent variable) to maximize the benefit of the suspicious system. Facing this challenge, we can only employ the undesirable exhaustive search over both the independent and dependent variables to achieve the optimal solution. To decrease the complexity, we further derive a tight approximation of the Lambert function and then develop easy-to-implement algorithms to find the sub-optimal solutions. Numerical results demonstrate the effectiveness of our proposed strategies compared to conventional passive eavesdropping.