In arXiv:1904.12783, we showed through numerical simulations of a minimally coupled massive Klein-Gordon scalar field that it is possible to grow "hair" on a Schwarzschild black hole if one assumes a periodically time-varying (but spatially homogeneous) background. We saw a non-trivial profile emerge on a timescale related to the mass of the black hole, both with and without backreaction of the field on the metric. The results are particularly relevant for scalar-tensor theories of gravity and dark matter models consisting of a massive scalar, e.g. axions. Our simulations are also a first step in studying the impact of such a background on a binary merger, potentially resulting in a imprint on the gravitational wave merger signal.