Probably the most well-known solution to automated test case minimization is the minimizing Delta Debugging algorithm (DDMIN). It is widely used because it “just works” on any kind of input. In this paper, we focus on the fixed-point iteration of DDMIN (named DDMIN*), more specifically whether it can improve on the result of the original algorithm. We present a carefully crafted example where the output of DDMIN could be reduced further, and iterating the algorithm finds a new, smaller local optimum. Then, we evaluate the idea on a publicly available test suite. We have found that the output of DDMIN* was usually smaller than the output of DDMIN. Using characters as units of reduction, the output became smaller by 67.94% on average, and in the best case, fixed-point iteration could improve as much as 89.68% on the output size of the original algorithm.