It is known that for a gauge-invariant formulation of QED and gravity, one should dress particles with Wilson lines stretching out to infinity. When considering asymptotic particles of scattering processes, such dressed particle states reduce to the infrared-finite states of Faddeev and Kulish. In quantum field theories in flat spacetime, the dressings of asymptotic states are known to carry a definite leading soft charge of the asymptotic symmetry, which can be interpreted as soft hair at infinity. Some recent attempts to extend this to subleading order will be briefly mentioned. We explore how this analysis can be extended to curved spacetimes with boundary, in particular, the Rindler and Schwarzschild spacetimes. More specifically, we will show that infalling dressed matter on the future Rindler and Schwarzschild horizons implant soft horizon hair.