Environmental monitoring applications use energy harvesting to cover wide-range deployment, where devices are powered by ambient energy and operate intermittently when energy is sufficient. In such an intermittent networked system (NIS), a sink node is used to forward the environmental data collected by sensors to a central controller to reflect the physical environment status. Nevertheless, existing data forwarding algorithms for NISs cannot fulfill modern application requirements, where multiple types of data with different timeliness requirements (i.e., multipriorities) are desired to report real-time environmental data for monitoring critical situations. Without considering the multipriorities, we show in this article that it introduces a new problem: unevenness of data freshness. We then propose the sink node-based evenness-aware update forwarding (iSAFE) algorithm to provide evenness among different priorities of data sources in NISs. iSAFE consists of three important components: 1) a theoretical analysis to derive the optimal data forwarding interval between two adjacent status updates from the sensor; 2) an evenness-aware forwarding algorithm to adaptively adjust the forwarding interval based on the runtime status; and 3) a fresh-aware energy preservation algorithm to maintain the freshness of collected data. The experimental results show that iSAFE can achieve up to 682% evenness (94.47% close to the ideal) and 53.3% data freshness compared to the state of the art while being energy-efficient and scalable, suitable for modern applications.