For the last two decades a large number of different automatic modulation classification (AMC) algorithms were developed, and many improvements in classification performance are reported. This was commonly achieved by engaging complex structures of neural networks, or other adaptable mechanisms for achieving better precision, when it comes to decisionmaking. Still, from practical implementation point of view, low algorithm complexity, economical usage of resources and fast execution remain to represent very desirable properties of an AMC algorithm. These properties are recognized in AMC algorithms based on higher-order cumulants as classification features, so their further improvement is of interest. Previous performance analysis of an algorithm based on sixthorder cumulants, in scenarios with complex valued signals' classification, showed that improvements are possible in the context of resources engaged and speed of execution. In this paper a novel approach is presented, for improving the correctness of classification process with sixthorder cumulants and simple twostep feature extraction structure, by engaging a new method for reduction of observed signal's modulation order which directly improves the classification performance. While tested with sixthorder cumulants, proposed method preserves good statistical properties of signal's higher-order cumulants in general, so it can be adopted in other AMC algorithms as well. Proposed modulation order reduction method is described in details, tested through computer simulations within the sixthorder cumulant AMC algorithm, and achieved improvements in performance are presented and explained.