Recently a mechanism called stagnation detection was proposed that automatically adjusts the mutation rate of evolutionary algorithms when they encounter local optima. The so-called SD-(11) EA introduced by Rajabi and Witt (2022) adds stagnation detection to the classical (11) EA with standard bit mutation. This algorithm flips each bit independently with some mutation rate, and stagnation detection raises the rate when the algorithm is likely to have encountered a local optimum. In this article, we investigate stagnation detection in the context of the -bit flip operator of randomized local search that flips bits chosen uniformly at random and let stagnation detection adjust the parameter . We obtain improved runtime results compared with the SD-(11) EA amounting to a speedup of at least , where is the so-called gap size, that is, the distance to the next improvement. Moreover, we propose additional schemes that prevent infinite optimization times even if the algorithm misses a working choice of due to unlucky events. Finally, we present an example where standard bit mutation still outperforms the -bit flip operator with stagnation detection.