  In a previous paper we have considered the Regge-Wheeler equation for fields
of spin $s=0$, $1$ or $2$ on the Schwarzschild spacetime in coordinates that
are regular at the horizon. In particular, we have constructed in
Eddington-Finkelstein coordinates exact solutions in terms of series that are
regular at the horizon and converge on the entire open domain from the central
singularity to infinity. Here we extend this earlier work in two different
directions. Firstly, we consider in Eddington-Finkelstein coordinates a massive
scalar field that can serve as a dark matter candidate. Secondly, we extend the
treatment of the massless case to Painlev\'e-Gullstrand coordinates, which are
associated with radially infalling observers.
