Context: Software testing based on finite-state machines. Objective:Improving the performance of existing testing methods by construction of more efficient separating sequences, so that states entered by a system under test can be identified in a much shorter span of time. Method: This paper proposes an efficient way to construct separating sequences for subsets of states for any deterministic finite-state machine. It extends an existing algorithm that builds an adaptive distinguishing sequence (ADS) from a splitting tree to machines that do not possess an ADS. Our extension to this construction algorithm allows one not only to construct a separating sequence for any subset of states but also form sets of separating sequences, such as harmonized state identifiers (HSI) and incomplete adaptive distinguishing sequences, that are used by efficient testing and learning algorithms. Results: The experiments confirm that the length and number of test sequences produced by testing methods that use HSIs constructed by our extension is significantly improved. Conclusion:By constructing more efficient separating sequences the performance of existing test methods significantly improves.