Security-constrained optimal power flow (SCOPF) and security-constrained unit commitment (SCUC) problems are necessary tools for system operators for operational planning and near to real-time operation. The solution time of these problems are challenging mainly due to their inherent large size. Previous studies have shown that relatively few of those problems' constraints serve to enclose their feasible set of solutions. Therefore, the constraints that do not contribute to the feasible set of solutions could be discarded to decrease the size of these problems and their associated solution times. Umbrella constraint discovery (UCD) has been proposed to identify and rule out redundant constraints in dc-SCOPF problems. In this paper, we propose an improvement over the original UCD formulation that exploits the structure of its parent SCOPF problem. This new partial UCD approach can lead to significant speed-ups in terms of UCD solution time and size. Based on the encouraging results for partial UCD on SCOPF, we apply the technique on SCUC. Alike in SCOPF, partial UCD can efficiently strip out non-umbrella constraints off SCUC. We find, however, that because of its structure, SCUC has a much lower proportion of non-umbrella constraints.