Systems with long-range interactions display a short-time relaxation towards Quasi Stationary States (QSSs) whose lifetime increases with system size. The application of Lynden-Bells theory of violent relaxation to the Hamiltonian Mean Field model leads to the prediction of out-of-equilibrium first and second order phase transitions between homogeneous (zero magnetization) and inhomogeneous (non-zero magnetization) QSSs, as well as an interesting phenomenon of phase re-entrances. We compare these theoretical predictions with direct $N$-body numerical simulations. We confirm the existence of phase re-entrance in the typical parameter range predicted from Lynden-Bells theory, but also show that the picture is more complicated than initially thought. In particular, we exhibit the existence of secondary re-entrant phases: we find un-magnetized states in the theoretically magnetized region as well as persisting magnetized states in the theoretically unmagnetized region.
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