In Multi-access Edge Computing (MEC) environments, session continuity during user mobility remains a pressing challenge due to decentralized infrastructure and high-throughput, latency-sensitive applications. Existing mobility protocols often rely on stateful mechanisms or centralized control, leading to increased signaling overhead, limited scalability, and vulnerability to performance degradation in dynamic networks.This paper introduces the Server Search and Select Algorithm Protocol (SSSAP), a lightweight, UDP-based handover protocol tailored for MEC deployments. The protocol is an extension of our previous work on a handover Server Search and Selection Algorithm (SSSA). SSSAP enables seamless session redirection through a three-phase signaling scheme (pre-handover, handover initiation, and handover termination), preserving service continuity without coupling session state to transport layers. The protocol's design features extensible headers for multi-metric evaluation and future security adaptation while maintaining minimal dependency on intermediary control nodes. Through extensive simulation and testing, we have validated the SSSAP efficiency across user equipment nodes and MEC servers. Results demonstrate high handover success rates, low-session setup delays, and balanced server load distribution. SSSAP achieves superior performance in mobility robustness, packet loss mitigation, and integration simplicity. The research outcomes position SSSAP as a scalable and application-agnostic mobility protocol for MEC systems, especially in vehicular and high-mobility scenarios.