The real-time monitoring and autonomous decision making through unmanned aerial vehicles (UAVs) are the potential applications of future networks. Vertical handover in future networks is a mechanism to switch communication between different network access technologies like Wireless Local Area Network (WLAN), Worldwide Interoperability for Wireless Microwave Access (WiMAX), Third-Generation (3G), Fourth-Generation (4G), and Fifth-Generation (5G) mobile technologies. These technologies have significant importance in providing fast, reliable, and timely communication. However, during a vertical handover, an inadequate delay and packet loss can cause considerable disruption in maintaining communication sessions and results in intolerable end-to-end delay, disconnectivity, and poor packet delivery ratio. The proposed work addresses the vertical handover method in UAVs communication by designing a relay-based vertical handover technique. The relay UAVs is an assistant node, requiring an organized and intelligent deployment that assists in vertical handover and communication by minimizing the average packet loss and average delay from source to destination. Moreover, a multicriteria handover parameter triggering is used for seamless and more extended network coverage. Extensive simulations using S-shaped and U-shaped trajectories are designed and simulated for relay-based vertical handover performance evaluation. The results obtained show that our proposed relay-based handover method offers seamless connectivity and high-performance experienced during the vertical handover process. The extensive comparison with state-of-the-art techniques proves that the proposed method is better in terms of 18% handover success rate, 21% end-to-end delay, and 29% packet loss.