  The decay timescales of the quasinormal modes of a massive scalar field have
an intriguing behavior: they either grow or decay with increasing angular
harmonic numbers $\ell$, depending on whether the mass of the scalar field is
small or large. We identify the properties of the effective potential of the
scalar field that leads to this behavior and characterize it in detail. If the
scalar field is non-minimally coupled, considered here, the scalar quasinormal
modes will leak into the gravitational wave signal and will have decaying times
that are comparable or smaller than those typical in General Relativity. Hence,
these modes could be detectable in the future. Finally, we find that the
anomalous behavior in the decay timescales of quasinormal modes is present in a
much larger class of models beyond a simple massive scalar field.
