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In this paper, we define modules with the properties (delta-GSCE) and (delta-GSCEE) by adapting Zoschinger's modules with the properties (E) and (EE) and we investigate the structure of modules with these properties. It is shown that: (1) a module has the property (delta-GSCEE) iff every submodule has the property (delta-GSCE); (2) the property (delta-GSCE) is inherited by direct summands; (3) for an R-module over a delta-V-ring, M has the property (delta-GSCE) iff M is cofinitely injective; (4) if R is a delta-semiperfect ring, then every left R-module has the property (delta-GSCE).