Hampton introduced an interesting type of central configuration, stacked central configurations, in 2005. He also raised two open questions about them. The goal of this work is to deepen our understanding of the stacked central configurations by providing a complete classification for the case that the number of the added bodies is one. Two open questions are also answered completely. In particular, we obtain a general result on the geometric property of the co-circular n-body central configurations, which is important to the understanding of the co-circular central configurations.