For a closed system at equilibrium the entropy is maximum. Is this a local maximum or is it a global maximum?
To allow a meaningful answer, it would be necessary to qualify the maximum. Maximum with respect to which variable? In thermodynamics the correct (and meaningful) statement is "maximum with respect to the variables (different from the thermodynamic state variables describing the isolated system) which represent the entropy dependence on all possible internal constraints" (i.e. constraints in the isolated system).
From this principle, i.e. from this sentence which condensate a long series of experiences, it is possible to obtain many consequences, like the equilibrium conditions or even the condition of concavity of the entropy as function of the variables which describe the macroscopic state of the isolated system, which, I stress, are not the same which describe the constraints.
So, from the maximum principle, one can get the concavity of entropy with respect to the state variables, by carefully choosing the kind of constraint.
However, such concavity as function of the state variables, does not imply strict concavity, or even concavity of entropy with respect to any possible internal constraint. For example, one could think of a constraint forcing an atomic system to stay only in two ordered crystalline structures (maybe not easy in a lab but not complicate in a computer simulation). For such constrained system one could have local maxima, with the highest being the true stable state and the remaining one, being a metastable system.
Probably the most interesting question could be: if we remove all the internal constraints, how can we know if there is a unique final equilibrium state?
And maybe this was the intended original question. Well, at the best of my knowledge, there is no definite answer. And there is a good reason for that. It is possible to imagine systems which do not reach equilibrium at all (non ergodic systems). Thus, I would consider the request of a unique maximum value of the entropy as an additional request for thermodynamic systems.