Earlier work used partial invariance to identify regions in state space where feedback linearization can be used despite the presence of unstable inverse dynamics. Such regions can be used as terminal regions in MPC with obvious advantages. Considering SISO bilinear systems, this paper exploits the fact that feedback linearization steers the state to the kernel of the output map. By restricting attention to this kernel, the paper develops results allowing for significant enlargement of the terminal region. Expressions are also given for maximal partially invariant sets, the recursive use of which leads to significant further enlargement.