  We propose an adaptive stencil construction for high order accurate finite
volume schemes aposteriori stabilized devoted to solve one-dimensional
steady-state hyperbolic equations. High-accuracy (up to the sixth-order
presently) is achieved thanks to polynomial reconstructions while stability is
provided with an aposteriori MOOD method which controls the cell polynomial
degree for eliminating non-physical oscillations in the vicinity of
discontinuities. We supplemented this scheme with a stencil construction
allowing to reduce even further the numerical dissipation. The stencil is
shifted away from troubles (shocks, discontinuities, etc.) leading to less
oscillating polynomial reconstructions. Experimented on linear, B\"urgers', and
Euler equations, we demonstrate that the adaptive stencil technique manages to
retrieve smooth solutions with optimal order of accuracy but also irregular
ones without spurious oscillations. Moreover we numerically show that the
approach allows to reduce the dissipation still maintaining the essentially
non-oscillatory behavior.
