  We give effective bilipschitz bounds on the change in metric between thick
parts of a cusped hyperbolic 3-manifold and its long Dehn fillings. In the thin
parts of the manifold, we give effective bounds on the change in complex length
of a short closed geodesic. These results quantify the filling theorem of Brock
and Bromberg, and extend previous results of the authors from finite volume
hyperbolic 3-manifolds to any tame hyperbolic 3-manifold. To prove the main
results, we assemble tools from Kleinian group theory into a template for
transferring theorems about finite-volume manifolds into theorems about
infinite-volume manifolds. We also prove and apply an infinite-volume version
of the 6-Theorem.
