  Subregular W-algebras are an interesting and increasingly important class of
quantum hamiltonian reductions of affine vertex algebras. Here, we show that
the $\mathfrak{sl}_{n+1}$ subregular W-algebra can be realised in terms of the
$\mathfrak{sl}_{n+1}$ regular W-algebra and the half lattice vertex algebra
$\Pi$. This generalises the realisations found for $n=1$ and $2$ in
[arXiv:1711.11342, arXiv:2007.00396] and can be interpreted as an inverse
quantum hamiltonian reduction in the sense of Adamovi\'c. We use this
realisation to explore the representation theory of $\mathfrak{sl}_{n+1}$
subregular W-algebras. Much of the structure encountered for
$\mathfrak{sl}_{2}$ and $\mathfrak{sl}_{3}$ is also present for
$\mathfrak{sl}_{n+1}$. Particularly, the simple $\mathfrak{sl}_{n+1}$
subregular W-algebra at nondegenerate admissible levels can be realised purely
in terms of the $\mathsf{W}_{n+1}$ minimal model vertex algebra and $\Pi$.
