Copper clusters experience a shape varation from layered to compact at the aggregation number of N = 16. Based on an extensive search for the structures of the low-lying neutral and charged isomers of Cu16 clusters, we address in this study the challenge in a structure search for clusters at shape-variation sizes, which arises from the structural diversity of the low-lying isomers. In order to reduce the bias on the structures with different shapes, a multi-step approach with a large number of candidates is applied to screen the structure pools which are necessarily larger than usual structure search of other sizes. In addition to the previously reported layered and compact structures, a third kind of structure, which can be recognized as a hybrid of layered and compact structures, is identified. Moreover, one of these hybrid structures is predicted to be most stable among the isomers. These hybrid structures not only bridge the structural and electronic properties of the distinct layered and compact structures, but also help understand the growth pattern of copper clusters. A set of new structures of anionic and cationic Cu16 isomers is also presented, starting from the newly established structure pool. The computed properties based on the identified neutral and ionic ground-state structures agree well with the available experimental data.