We formulate and prove examples of a conjecture which describes the \(\mathcal{W}-\text{algebras}\) in type A as successive quantum Hamiltonian reductions of affine vertex algebras associated with several hook-type nilpotent orbits. This implies that the affine coset subalgebras of hook-type \(\mathcal{W}-\text{algebras}\) are building blocks of the \(\mathcal{W}-\text{algebras}\) in type A. In the rational case, it turns out that the building blocks for the simple quotients are provided by the minimal series of the regular \(\mathcal{W}-\text{algebras}\) . In contrast, they are provided by singlet-type extensions of \(\mathcal{W}-\text{algebras}\) at collapsing levels which are irrational. In the latter case, several new sporadic isomorphisms between different \(\mathcal{W}-\text{algebras}\) are established.
