  The physical interpretation and eventual fate of gravitational singularities
in a theory surpassing classical general relativity are puzzling questions that
have generated a great deal of interest among various quantum gravity
approaches. In the context of loop quantum gravity (LQG), one of the major
candidates for a non-perturbative background-independent quantisation of
general relativity, considerable effort has been devoted to construct effective
models in which these questions can be studied. In these models, classical
singularities are replaced by a "bounce" induced by quantum geometry
corrections. Undesirable features may arise however depending on the details of
the model. In this paper, we focus on Schwarzschild black holes and propose a
new effective quantum theory based on polymerisation of new canonical phase
space variables inspired by those successful in loop quantum cosmology. The
quantum corrected spacetime resulting from the solutions of the effective
dynamics is characterised by infinitely many pairs of trapped and anti-trapped
regions connected via a space-like transition surface replacing the central
singularity. Quantum effects become relevant at a unique mass independent
curvature scale, while they become negligible in the low curvature region near
the horizon. The effective quantum metric describes also the exterior regions
and asymptotically classical Schwarzschild geometry is recovered. We however
find that physically acceptable solutions require us to select a certain subset
of initial conditions, corresponding to a specific mass (de-)amplification
after the bounce. We also sketch the corresponding quantum theory and
explicitly compute the kernel of the Hamiltonian constraint operator.
