In 2021, Dall'Ara and Mongodi introduced the Levi core $\mathfrak{C}(\mathcal{N})$, a subdistribution of the Levi null space, and showed that it could be used to study regularity properties of the $\overline{\partial}$-Neumann operator. In 2022, the speaker showed that if the support of the Levi core satisfies Property ($P$), then the boundary of the domain satisfies Property ($P$). He also gave an example of a Hartogs domain in $\mathbb{C}^2$ which satisfies Property ($P$) yet the support of the Levi core was large in a measure-theoretic sense. In this talk, we modify the algorithm for constructing the Levi core to give a new modified core $\mathcal{M}\mathfrak{C}$. This new core is a subdistribution of the Levi core $\mathfrak{C}(\mathcal{N})$. We examine properties of the modified Levi core in relation to the regularity of the $\overline{\partial}$-Neumann operator and revisit the Hartogs domain example. This is joint work with Tanuj Gupta (Texas A\&M University) and Emil Straube (Texas A\&M University).