Is every theory meeting Gödels incompleteness, also incomplete below its consistency level?

Question

Is it always the case that for any theory $T$ that meets Godel's criteria for incompleteness, there is a sentence $P$ such that neither $T \vdash P$, nor $T\vdash \neg P$; and such that $T+P$ is equi-consistent with $T$, and $T+ \neg P$ is equi-consistent with $T$?