President Obama is facing another immigration dilemma exacerbated by his efforts to spare up to 5 million illegal immigrants from deportation: how to address the roughly 6 million who were not covered by his executive action.

Soon after he announced his unilateral move, the most sweeping overhaul to the immigration system in decades, Obama heard complaints from immigration rights advocates that he hadn’t gone far enough. They urged him to grant protections to millions of additional illegal immigrants.