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We present methods for computing the explicit decomposition of the minimalsimple affine $W$-algebra $W_k(\mathfrak g, \theta)$ at a conformal level $k$as a module for its maximal affine subalgebra $\mathcal V_k(\mathfrakg^{\natural})$. A particular emphasis is given on the application of affinefusion rules to the determination of branching rules. In almost all cases when$\mathfrak g^{\natural}$ is a semisimple Lie algebra, we show that, for asuitable conformal level $k$, $W_k(\mathfrak g, \theta)$ is isomorphic to anextension of $\mathcal V_k(\mathfrak g^{\natural})$ by its simple module. Weare able to prove that in certain cases $W_k(\mathfrak g, \theta)$ is a simplecurrent extension of $\mathcal V_k(\mathfrak g^{\natural})$. In order toanalyze more complicated non simple current extensions at conformal levels, wepresent an explicit realization of the simple $W$-algebra $W_{k}(sl(4),\theta)$ at $k=-8/3$. We prove, as conjectured in arXiv:1407.1527, that$W_{k}(sl(4), \theta)$ is isomorphic to the vertex algebra $\mathcal R^{(3)}$,and construct infinitely many singular vectors using screening operators. Wealso construct a new family of simple current modules for the vertex algebra$V_k (sl(n))$ at certain admissible levels and for $V_k (sl(m | n)), m\ne n,m,n\geq 1$ at arbitrary levels.

Conformal embeddings of affine vertex algebras in minimal $W$-algebras II: decompositions