Recently, intelligent reflecting surface (IRS) and unmanned aerial vehicle (UAV) have been introduced into wireless communication systems to enhance the performance of air-ground transmission. To make a good balance between performance, cost, and power consumption, a hybrid-IRS-and-UAV-assisted directional modulation (DM) network is investigated in this paper, where the hybrid IRS consists of passive and active reflecting elements. To maximize the achievable rate, three optimization algorithms, called maximum signal-to-noise ratio (SNR)-fractional programming (FP) (Max-SNR-FP), maximum SNR-equal amplitude reflecting (EAR) (Max-SNR-EAR), and maximum SNR-majorization-minimization (MM) (Max-SNR-MM), are proposed to jointly design the beamforming vector and phase shift matrix (PSM) of hybrid IRS by alternately optimizing one and giving another. The Max-SNR-FP method employs the successive convex approximation and FP methods to derive the beamforming vector and hybrid IRS PSM. The Max-SNR-EAR method adopts the maximum signal-to-leakagenoise ratio method and the criteria of phase alignment and EAR to design them. In addition, the Max-SNR-MM method utilizes the MM criterion to derive the IRS PSM. Simulation results show that the rates harvested by the proposed three methods are slightly lower than those of active IRS with higher power consumption, which are 35 percent higher than those of no IRS and random phase IRS, while passive IRS achieves only about 17 percent rate gain over the latter. Moreover, compared to Max-SNR-FP, the proposed Max-SNR-EAR and Max-SNR-MM methods make an obvious complexity degradation at the price of a slight performance loss.