In this paper, the Allen-Cahn model equation has 0 and 1 as solutions and as minima of the associated functional. The existence of solutions that are local minima and are close to 0 and 1 on a prescribed pattern of sets has been established in [Calc. Var. Partial Differential Equations 47 (2013), 1-23]. Here related solutions of higher topological type are obtained using minimax and heat flow arguments on a family of invariant sets.