Vehicular Ad-hoc Networks (VANETs) are highly dynamic and self-organized ad hoc wireless networks that enable vehicles to set up and maintain communication without any central base station. In multi-hop-based VANETs, messages are disseminated throughout the network through a relaying. However, the dissemination of time-critical emergency warning messages remains challenging due to the broadcast storm problem. The widely proposed broadcast suppression strategies are either beacon-based, which demands extra bandwidth or do not scale well under different traffic conditions. Thus, this paper proposes a multi-criteria-based broadcast suppression scheme for multi-hop VANETS. The proposed scheme utilizes multiple node attributes, including relative velocity, traffic density, lane weight (moving lane), link stability, movement angle and distance to sender, to estimate a node's quality. Upon receiving a new message, each node can contend for the next-hop relay position using its quality value such that a high-quality node is granted the earliest channel access to broadcast the data packet. The performance of the proposed scheme is validated using data generated from traffic simulation in the OMNET++ environment under different traffic conditions. The results from the simulation claim that the proposed strategy alleviates broadcast overhead by 30% and significantly improves the overall dissemination efficiency by 48.97% compared to the bench-marked methods. The results showed that the objectives are met by significantly improving dissemination efficiency with increased reachability, reduced broadcast overhead and end-to-end delay. Moreover, the scheme enables each receiving node to evaluate its broadcasting suitability autonomously, making the relay selection process dynamic and adaptive to any traffic condition.