  We compute the adjoint twisted Reidemeister torsion for closed oriented
hyperbolic $3$-manifolds and for hyperbolic $3$-manifolds with toroidal
boundary. In our formula, we consider the manifold as obtained by doing a
Dehn-filling along suitable boundary components of a fundamental shadow link
complement, and the formula is in terms of the logarithmic holonomy of the
meridians of the boundary components. As an important special case, we also
write down a formula of the adjoint twisted Reidemeister torsion for the double
of a hyperbolic $3$-manifold with totally geodesic boundary in terms of the
edge lengths of a geometric ideal triangulation of the manifold. These
unexpected formulas were inspired by, and played an important role in, the
study of the asymptotic expansion of quantum invariants\,\cite{WY}.
