  This paper generalizes the results obtained by the authors in
\cite{dangHomogenizationNondiluteSuspension2021} concerning the homogenization
of a non-dilute suspension of magnetic particles in a viscous flow. More
specifically, in this paper, a restrictive assumption on the coefficients of
the coupled equation, made in \cite{dangHomogenizationNondiluteSuspension2021},
that significantly narrowed the applicability of the homogenization results
obtained, is relaxed and a new regularity of the solution of the fine-scale
problem is proven. In particular, we obtain a global $L^{\infty}$-bound for the
gradient of the solution of the scalar equation $-\mathrm{div} \left[
\mathbf{a} \left( x/\varepsilon \right)\nabla \varphi^{\varepsilon}(x) \right]
= f(x)$, uniform with respect to microstructure scale parameter $\varepsilon\ll
1$ in a small interval $(0,\varepsilon_0)$, where the coefficient $\mathbf{a}$
is only \emph{piecewise} H\"{o}lder continuous. Thenceforth, this regularity is
used in the derivation of the effective response of the given suspension
discussed in \cite{dangHomogenizationNondiluteSuspension2021}.
