To address the issue of inaccurate performance evaluation caused by approximating the communication channel model through line-of-sight links in existing covert communication methods, and to better adapt to real-world communication channel conditions, a novel covert communication model assisted by an unmanned aerial vehicle (UAV) equipped with a multifunctional reconfigurable intelligent surface is proposed. Specifically, based on a hybrid link of line-of-sight and non-line-of-sight, the model utilizes the multifunctional reconfigurable intelligent surface as a flexible relay and optimizes the UAV′s flying location to enhance the signal-to-noise ratio and effective throughput. The hybrid channel model expression is first derived. Then, the covert constraint is transformed from an incomplete gamma function to a constraint related to relative entropy, and the UAV′s transmission power and block length are correspondingly optimized. Finally, by analyzing the functional characteristics of SNR and effective throughput under the relative entropy constraints, the optimal flying location for the UAV is obtained. Compared with non-line-of-sight link channels, the proposed model improved the SNR by 42% at the optimal flight position. Extensive numerical results indicate that the proposed model outperforms existing works in enhancing SNR and effective throughput.