The set covering problem (SCP) is an NP-complete optimization problem, fitting with many problems in engineering. The traditional SCP formulation does not directly address both solution unsatisfiability and set redundancy aspects. As a result, the solving methods have to control these aspects to avoid getting unfeasible and nonoptimized in cost solutions. In the last years, an alternative SCP formulation was proposed, directly covering both aspects. This alternative formulation received limited attention because managing both aspects is considered straightforward at this time. This paper questions whether there is some advantage in the alternative formulation, beyond addressing the two issues. Thus, two studies based on a metaheuristic approach are proposed to identify if there is any concept in the alternative formulation, which could be considered for enhancing a solving method considering the traditional SCP formulation. As a result, the authors conclude that there are concepts from the alternative formulation, which could be applied for guiding the search process and for designing heuristic feasibilit\y operators. Thus, such concepts could be recommended for designing state-of-the-art algorithms addressing the traditional SCP formulation.