The modular product and existential closure II

In this article we study the modular graph product, ♦, that is known to preserve the property of being 3-existentially closed (i.e., 3-e.c.). We produce new families of 3-e.c. graphs G♦H such that neither G nor H is required to be 3-e.c. Assuming that G is weakly 3-existentially closed with certain adjacency properties, we find the sufficient conditions on the adjacency properties of H such that G♦H is 3-e.c. The graph G can have as few as four vertices, and it is settled in this article that H can have as few as 24 vertices. These altogether present an improvement in comparison to when at least one of G or H were required to be 3-e.c.