Author’s abstract: This article is a continuation of a recent paper by the author and R. Z. Buzyakova. New results are obtained with respect to the next natural question: how complex can a space be that is the union of two (of a finite family) “nice” subspaces? Our approach is based on the notion of a \(D\)-space introduced by E. van Douwen and on a generalization of this notion, the notion of \(aD\)-space. It is proved that if a space \(X\) is the union of a finite family of subparacompact subspaces, then \(X\) is an \(aD\)-space. Under (CH), it follows that if a separable normal \(T_1\)-space \(X\) is the union of a finite number of subparacompact subspaces, then \(X\) is Lindelöf. It is also established that if a regular space \(X\) is the union of a finite family of subspaces with a point-countable base, then \(X\) is a \(D\)-space. Finally, a certain structure theorem for unions of finite families of spaces with a point-countable base is established, and numerous corollaries are derived from it. Also, many new open problems are formulated.