Refreshing algorithms are a critical ingredient for secure masking. They are instrumental in enabling sound composability propertiesfor complex circuits, and their randomness requirements dominate theperformance overheads in (very) high-order masking. In this paper, we improve a proposal of mask refreshing algorithms from EUROCRYPT2017, that has excellent implementation properties in software and hardware, in two main directions. First, we provide a generic proof that this algorithm is secure at arbitrary orders – a problem that was left open so far. We introduce Parametrized Non-Interference as a new technical ingredient for this purpose, that may be of independent interest. Second, we use automated tools to further explore the design space of such algorithms and provide the best known parallel mask refreshing gadgets for concretely relevant security orders. Incidentally, we also prove the security of a recent proposal of mask refreshing with improved resistance against horizontal attacks from CHES 2017.