Second-order abstract syntax (SOAS) provides a well-founded framework when working with languages that have arbitrary binders. E-unification for SOAS is powerful enough to encode both E-unification and higher-order unification (HOU). In a previous work, we have presented an E-unification procedure, drawing inspiration from both E-unification and HOU algorithms. In this paper, we present a work-in-progress on an more efficient version of the procedure, aiming to bring E-unification for SOAS to perform on par with HOU algorithms. More specifically, in this work we focus on a generalisation of Huet-style projections (also known as partial bindings in HOU literature), commonly used in HOU algorithms.