A cross-bifix-free code of lengthnover Zqis a non-empty subset of Znqsuch that the prefix set of each codeword is disjoint from the suffix set of every codeword. To achieve good performance in communication systems, it is desirable to construct cross-bifix-free codes with large size. Recently, Wang and Wang generalized the classical cross-bifix-free codes presented by Levenshtein, Gilbert and Cheeet al. by constructing a new family of cross-bifix-free codesS(k)I,J(n). The codeS(k)I,J(n) is nearly optimal in terms of its size and non-expandable ifk=n- 1 or 1 ≤kn/2. There are three major ingredients in this paper. The first is to improve the results in [Cheeet al., IEEE-TIT, 2013] and [Wang and Wang, IEEE-TIT, 2022] in which we prove that the codeS(k)I,J(n) is non-expandable if and only ifk=n- 1 or 1 ≤kn/2. The second ingredient contributes to a new family of cross-bifix-free codesU(t)I,J(n). This new code enables us to construct non-expandable cross-bifix-free codesS(k)I,J(n) ᑌU(t)I,J(n) wheneverS(k)I,J(n) is expandable. The union ofU(t)I,J(n) andS(k)I,J(n) enlarges the size ofS(k)I,J(n). Finally, we give an explicit formula for the size ofS(k)I,J(n) ᑌU(t)I,J(n).