  Recently, Ursula Hamenst\"adt and the author proved a stability result for
finite volume hyperbolic metrics in dimension three that does not assume any
upper volume bounds, but that requires an exponentially fine control of the
metric in the thin part of the manifold. We use a bootstrap argument to extend
the result allowing for a weaker exponential control of the metric. This is
achieved by formulating an abstract axiomatic framework.
