  Originating in the work of A.M. Semikhatov and D. Adamovi\'c, inverse
reductions are embeddings involving W-algebras corresponding to the same Lie
algebra but different nilpotent orbits. Here, we show that an inverse reduction
embedding between the affine $\mathfrak{sl}_{n+1}$ vertex operator algebra and
the minimal $\mathfrak{sl}_{n+1}$ W-algebra exists. This generalises the
realisations for $n=1,2$ in [arXiv:1711.11342, arXiv:2110.15203]. A similar
argument is then used to show that inverse reduction embeddings exists between
all hook-type $\mathfrak{sl}_{n+1}$ W-algebras, which includes the
principal/regular, subregular, minimal $\mathfrak{sl}_{n+1}$ W-algebras, and
the affine $\mathfrak{sl}_{n+1}$ vertex operator algebra. This generalises the
regular-to-subregular inverse reduction of [arXiv:2111.05536], and similarly
uses free-field realisations and their associated screening operators.
