The aim of this paper is to introduce and study new types of strong weakly $\mu$-compact spaces in generalized topological spaces with a hereditary class, called weakly $\mathcal{S}\mu\mathcal{H}$-compact and weakly $\mathbf{S}-\mathcal{S}\mu\mathcal{H}$-compact spaces. Some fundamental properties of these spaces are given. Also, we investigate the invariants of weakly $\mathcal{S}\mu\mathcal{H}$-compact and weakly $\mathbf{S}-\mathcal{S}\mu\mathcal{H}$-compact spaces under functions.