  This paper is the third in a series of four papers aiming to describe the
(almost integral) Chow ring of $\overline{\mathcal{M}}_3$, the moduli stack of
stable curves of genus $3$. In this paper, we compute the Chow ring of
$\widetilde{\mathcal{M}}_3^7$ with $\mathbb{Z}[1/6]$-coefficients.
