Simulated Annealing (SA) is commonly considered as an efficient method to construct Maximin Latin Hypercube Designs (LHDs) which are widely employed for Experimental Design. The Maximin LHD construction problem may be generalized to the Maximin LHD completion problem in an instance of which the measurements have already been taken at certain points. The construction may then be seen as a particular case of completion with no points given in advance. As the Maximin LHD completion was proved NP-complete and inapproximable with a constant factor, the choice of SA to treat it shows itself naturally. The SA performance varies greatly depending on the mutation used. The completion problem is difficult because its nature changes when the number of given points varies. For a few fixed points, the completion behaves similarly to the construction problem. In the opposite situation, numerous fixed points restrain the search space considerably and a different mutation is appropriate. A phase transition exists between these extreme cases. We thus provide SA with a mechanism which selects an appropriate mutation. Our approach is based on the observation that the choice of a mutation can be seen as a bandit problem. It has to cope with changes in the environment, which evolves together with the thermal descent. The results obtained prove that the bandit-driven SA adapts itself on the fly to the completion problem nature. We believe that other parametrized problems, where SA can be employed, may significantly benefit from the use of a decision-making algorithm which selects the appropriate mutation.