In this article, via establishing a new atomic characterization of the Musielak–Orlicz Hardy space $H^{\phi }\left({\mathbb{R}}^n\right)$ [which is essentially deduced from the known molecular characterization of $H^{\phi }\left({\mathbb{R}}^n\right)$] and some estimates on a new discrete Littlewood–Paley $g$-function and a Peetre-type maximal function, together with using the known intrinsic $g$-function characterization of $H^{\phi }\left({\mathbb{R}}^n\right)$, the authors obtain several equivalent characterizations of $H^{\phi }\left({\mathbb{R}}^n\right)$ in terms of wavelets, which extend the wavelet characterizations of both Orlicz–Hardy spaces and the weighted Hardy spaces, and are available to the typical and useful Musielak–Orlicz Hardy space $H^{log}\left({\mathbb{R}}^n\right)$. The novelty of this approach is that the new adapted atomic characterization of $H^{\phi }\left({\mathbb{R}}^n\right)$ compensates the inconvenience in applications of the supremum appearing in the original definition of atoms, which play crucial roles in the proof of the main theorem of this article and may have further potential applications.