Although the modern automatic test pattern generation (ATPG) tools can efficiently produce near-optimal test sets with high fault-coverage for a circuit-under-test, a diagnostic test set (DTS), which is needed for fault localization, is much more challenging to construct. The DTS is used to analyze the responses of failing chips during manufacturing test for the purpose of identifying the root cause of observed errors. In this paper, a novel technique for selecting a powerful DTS for stuck-at faults from a pool of ATPG detection vectors is proposed. Unlike existing methods, this technique does not use any diagnostic test generation, circuit modification, or miter-based approach. It constructs a combinatorial cover of the pool to determine a test set with high diagnostic coverage (DC). Two variants of the covering algorithm are proposed based on this technique. The experimental results on several combinational and scan-based benchmark circuits demonstrate the effectiveness of our method in terms of the size of the DTS, DC, and CPU time.