  Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell
\geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in
the round $3$-sphere a closed embedded minimal surface that has genus $k\ell
m^2(N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of
$O(4)$, where $D_n$ is the dihedral group of order $2n$. Each such surface
resembles the union of $N$ nested topological tori, all small perturbations of
a single Clifford torus $\mathbb{T}$, that have been connected by $k\ell m^2
(N-1)$ small catenoidal tunnels, with $k \ell m^2$ tunnels joining each pair of
neighboring tori. In the large-$m$ limit for fixed $N$, $k$, and $\ell$, the
corresponding surfaces converge to $\mathbb{T}$ counted with multiplicity $N$.
