In this paper, we investigate the secrecy performance of a dual-hop cooperative relaying network, where a source node communicates with a destination node via an amplify-and-forward relay node and via direct link in the presence of a passive eavesdropper. In this network, the eavesdropper exploits the advantages of both direct and relay links. We assume that the source, relay, and eavesdropper nodes are mobile, and the destination node is stationary. Moreover, we consider that the source-relay, relay-destination, and source-destination links experience Rayleigh fading, whereas source-eavesdropper and relay-eavesdropper links undergo double-Rayleigh fading. Examples of such scenario are encountered in vehicle-to-vehicle communication and/or vehicle-to-infrastructure communication under cooperative vehicular networks. In particular, we derive new tight closed-form expressions for the secrecy outage probability and ergodic secrecy capacity of the considered system under mixed fading environments. In addition, we present an effective secrecy diversity order within the main-to-eavesdropper ratio ranges of practical interest, and demonstrate that the secrecy diversity order of 2 can be achieved. Finally, the numerical and simulation results corroborate our analytical findings, and show the occurrence conditions for secrecy outage probability and ergodic secrecy capacity floors, and impacts of channel conditions and relay and eavesdropper positions on the system secrecy performance.