  We consider codimension $1$ area-minimizing $m$-dimensional currents $T$ mod
an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $\Sigma$ of the
Euclidean space. We prove a suitable excess-decay estimate towards the unique
tangent cone at every point $q\in \mathrm{spt} (T)\setminus \mathrm{spt}^p
(\partial T)$ where at least one such tangent cone is $Q$ copies of a single
plane. While an analogous decay statement was proved in arXiv:2111.11202 as a
corollary of a more general theory for stable varifolds, in our statement we
strive for the optimal dependence of the estimates upon the second fundamental
form of $\Sigma$. This technical improvement is in fact needed in
arXiv:2201.10204 to prove that the singular set of $T$ can be decomposed into a
$C^{1,\alpha}$ $(m-1)$-dimensional submanifold and an additional closed
remaining set of Hausdorff dimension at most $m-2$.
