Covering properties build the foundation of stability and sensitivity analysis of solutions to a generalized equation and more specific optimization-related stationarity and equilibrium problems. It has been well-understood that metric regularity of the mapping defining the generalized equation is a key to furnish Lipschitzian stability of the solution of interest. With this work, we want to leave the ties of metric regularity, and establish new covering results under a regularity concept introduced by H. Gfrerer, whose relation to notions of 2-regularity has been elaborated recently. Among other things, our findings target a precise description of the set with nonempty interior being covered. Comprehensive comparisons with existing results for constrained equations are included, as well as an application to coupled constraint systems, arising in practically relevant settings, including variational and optimality systems, or complementarity systems over convex cones.