  In 2003, Klainerman and Nicol\`o \cite{Kl-Ni} proved the stability of
Minkowski in the case of the exterior of an outgoing null cone. Relying on the
method used in \cite{Kl-Ni}, Caciotta and Nicol\`o \cite{Ca-Ni} proved the
stability of Kerr spacetime in external regions, i.e. outside an outgoing null
cone far away from the Kerr event horizon. In this paper, we give a new proof
of \cite{Ca-Ni}. Compared to \cite{Ca-Ni}, we reduce the number of derivatives
needed in the proof, simplify the treatment of the last slice, and provide a
unified treatment of the decay of initial data which contains in particular the
initial data considered by Klainerman and Szeftel in \cite{KS:main}. Also,
concerning the treatment of curvature estimates, similar to \cite{ShenMink}, we
replace the vectorfield method used in \cite{Kl-Ni,Ca-Ni} by $r^p$-weighted
estimates introduced by Dafermos and Rodnianski in \cite{Da-Ro}.
