  We show that the Schwarzschild-(A)dS black hole mechanics possesses a hidden
symmetry under the three-dimensional Poincar\'e group. This symmetry shows up
after having gauge-fixed the diffeomorphism invariance in the symmetry-reduced
homogeneous Einstein-$\Lambda$ model and stands as a physical symmetry of the
system. It dictates the geometry both in the black hole interior and exterior
regions, as well as beyond the cosmological horizon in the Schwarzschild-dS
case. It follows that one can associate a set of non-trivial conserved charges
to the Schwarzschild-(A)dS black hole which act in each causally disconnected
regions. In T-regions, they act on fields living on spacelike hypersurface of
constant time, while in the R-region, they act on time-like hypersurface of
constant radius. We find that while the expression of the charges depend
explicitly on the location of the hypersurface, the charge algebra remains the
same at any radius in R-regions (or time in T-regions). Finally, the analysis
of the Casimirs of the charge algebra reveals a new solution-generating map.
The sl$(2,\mathbb{R})$ Casimir is shown to generate a one-parameter family of
deformation of the black hole geometry labelled by the cosmological constant.
This gives rise to a new conformal bridge allowing one to continuously deform
the Schwarzschild-AdS geometry to the Schwarzschild and the Schwarzschild-dS
solutions.
