An hourglass \(\Gamma_0\) is the graph with degree sequence \(\{4,2,2,2,2\}\). In this paper, for integers \(j\geq i\geq 1\), the bull \(B_{i,j}\) is the graph obtained by attaching endvertices of two disjoint paths of lengths \(i,j\) to two vertices of a triangle. We show that every 3-connected \(\{K_{1,3},\Gamma_0,X\}\)-free graph, where \(X\in \{ B_{2,12},\,B_{4,10},\,B_{6,8}\}\), is Hamilton-connected. Moreover, we give an example to show the sharpness of our result, and complete the characterization of forbidden induced bulls implying Hamilton-connectedness of a 3-connected {claw, hourglass, bull}-free graph.