  Let $(A,\alpha)$ be a system consisting of a $C^*$-algebra $A$ and an
automorphism $\alpha$ of $A$. We describe the primitive ideal space of the
partial-isometric crossed product $A\times_{\alpha}^{\textrm{piso}}\mathbb{N}$
of the system by using its realization as a full corner of a classical crossed
product and applying some results of Williams and Echterhoff.
