Graph neural networks (GNNs) have become an indispensable tool for analyzing relational data. Classical GNNs are broadly classified into three variants: convolutional, attentional, and message-passing. While the standard message-passing variant is expressive, its typical pair-wise messages only consider the features of the center node and each neighboring node individually. This design fails to incorporate contextual information contained within the broader local neighborhood, potentially hindering its ability to learn complex relationships within the entire set of neighboring nodes. To address this limitation, this work first formalizes the concept of neighborhood-contextualization, rooted in a key property of the attentional variant. This then serves as the foundation for generalizing the message-passing variant to the proposed neighborhood-contextualized message-passing (NCMP) framework. To demonstrate its utility, a simple, practical, and efficient method to parametrize and operationalize NCMP is presented, leading to the development of the proposed Soft-Isomorphic Neighborhood-Contextualized Graph Convolution Network (SINC-GCN). Across a diverse set of synthetic and benchmark GNN datasets, SINC-GCN demonstrates competitive performance against baseline GNN models, highlighting its expressivity and efficiency. Notably, it also delivers substantial and statistically significant performance gains in graph property prediction tasks, further underscoring the distinctive utility of neighborhood-contextualization. Overall, the paper lays the foundation for the NCMP framework as a practical path toward enhancing the graph representational power of classical GNNs.