This paper continues to study the construction of uninorms on bounded lattices. First, we present some new constructions of uninorms on an arbitrary bounded lattice L with some additional constraints on the neutral element. In particular, we obtain two idempotent uninorms on bounded lattices. Then we demonstrate that our new construction methods are different from some existing methods for the construction of uninorms on bounded lattices. In addition, some illustrative examples for the new constructions of uninorms on bounded lattices are provided. In particular, the results presented here provide a partial answer to a problem (as proposed by F. Karaçal and R. Mesiar (2015) in [23]), which is “when L is a chain (in particular, when is the unit interval) then to any pair of a t-norm on and a t-conorm on there are related uninorms. For which bounded lattices a similar claim is valid?”.