  We show that the classical double copy relationship for Kerr-Schild
spacetimes can be dimensionally reduced to give a natural notion of the double
copy for Kaluza-Klein theory with gravity coupled to a gauge field and a
dilaton. Under dimensional reduction the Kerr-Schild (KS) ansatz becomes the
stringy Kerr-Schild (sKS) ansatz introduced by Wu. This ansatz captures many
Kaluza-Klein black hole solutions, including single-charge black holes arising
in both gauged and ungauged supergravity theories. We identify the single copy
gauge field and scalar field of an arbitrary sKS solution. We show that the
boost-reduction procedure for generating charged black hole solutions can be
generalized to any stationary KS seed, and used to combine the metric with the
zeroth and single copies of that seed into a single sKS solution. Furthermore,
we comment on multi-charge solutions that can be written in a multi-sKS form,
proposing a double copy interpretation involving multiple single copy sectors.
