  We formulate and prove examples of a conjecture which describes the
W-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 W-algebras are building
blocks of the W-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 W-algebras. In contrast, they are provided by singlet-type
extensions of W-algebras at collapsing levels which are irrational. In the
latter case, several new sporadic isomorphisms between different W-algebras are
established.
