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In this paper, we study on modules that have a weak (ample) δ-supplement in every extension which are adapted Zöschinger's modules with the properties (E) and (EE). It is shown that: (1) Direct summands of modules with the property δ-(CWE) have the property δ-(CWE); (2) For a module M, if every submodule of M has the property δ-(CWE) then so does M; (3) For a ring R, R is δ-semilocal iff every R-module has the property δ-(CWE); (4) Every factor module of a finitely generated module that has the property δ-(CWE) also has the property δ-(CWE) under a special condition; (5) Let M be a module and L be a submodule of M such that L <<#948; M. If the factor module M/L has the property δ-(CWE), then so does M; (6) On a semisimple module the concepts of modules that have the property δ-(CE) and δ-(CWE) coincide with each other. [ABSTRACT FROM AUTHOR]