In recent years, isogeometric analysis has been an active topic of research in numerical mathematics. Using higher regularity finite-dimensional spaces for the finite element method allows for better approximation power per degree of freedom (DOF). Additionally, this allows the domain to be more accurately imported from Computer-Aided Design (CAD) software, which results in a reduction/elimination of domain meshing errors. For this reason, B-splines are commonly used as basis functions. However, adaptive splines allow for significant savings when capturing solutions or approximating geometries with localised features. One such adaptive spline methodology that is popular for its simple implementation and attractive properties is that of Truncated Hierarchical B-splines (THB-splines). We have previously demonstrated that a local B\'ezier projector can be created for THB-splines in any dimension $n$. We accomplished this by constructing locally-defined macro elements on which the THB-splines are linearly independent and applying local-macro-element-wise projections. These local projections are then smoothed to create a global projection using a weighted average. In this work, we show how some of the assumptions on the locally refined meshes can be relaxed (e.g., that of mesh admissibility) and how this projector can be utilized in adaptive simulations.