Earlier work used the concept of partial invariance in order to identify regions in state space where feedback linearization can be used despite the presence of unstable internal dynamics. The use of such regions as terminal regions in MPC has obvious advantages. This paper recognizes that feedback linearization applied to SISO bilinear systems steers the state to the kernel of the output map. Correspondingly, by restricting attention to this kernel the paper develops results which allow for the significant enlargement of the relevant terminal regions. Expressions are also given for maximal partially invariant sets and this used recursively can lead to significant further enlargement.