  We construct vertex algebraic intertwining operators among certain
generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$ and
calculate the corresponding fusion rules. Additionally, we show that under some
conditions these intertwining operators descend to intertwining operators among
one generalized Verma module and two (generally non-standard) irreducible
modules. Our construction relies on the irreducibility of the maximal proper
submodules of generalized Verma modules appearing in the Garland-Lepowsky
resolutions of standard $\widehat{\mathfrak{sl}(2,\mathbb{C})}$-modules. We
prove this irreducibility using the composition factor multiplicities of
irreducible modules in Verma modules for symmetrizable Kac-Moody Lie algebras
of rank $2$, given by Rocha-Caridi and Wallach.
