  Let $\ell$ be a rational prime. Previously, abelian $\ell$-towers of
multigraphs were introduced which are analogous to
$\mathbb{Z}_{\ell}$-extensions of number fields. It was shown that for a
certain class of towers of bouquets, the growth of the $\ell$-part of the
number of spanning trees behaves in a predictable manner (analogous to a
well-known theorem of Iwasawa for $\mathbb{Z}_{\ell}$-extensions of number
fields). In this paper, we give a generalization to a broader class of regular
abelian $\ell$-towers of bouquets than was originally considered. To carry this
out, we observe that certain shifted Chebyshev polynomials are members of a
continuously parametrized family of power series with coefficients in
$\mathbb{Z}_{\ell}$ and then study the special value at $s=1$ of the
Artin-Ihara $L$-function $\ell$-adically.
