Though traditional isogeometric analysis employs B-splines and NURBS because they serve as the workhorse for computer-aided design (CAD), there is significant interest in the use of unstructured splines to mirror the strengths of NURBS while mitigating their shortcomings. Particularly, NURBS-based methods lack high continuity between separate tensor-product patches and lack the capacity to directly operate on higher-order partial differential equations. Previously, Almost C1 splines have been introduced as a simple biquadratic spline space with C1 continuity everywhere except at edges directly emanating from both internal and boundary extraordinary points. These splines have optimal convergence properties, and boast characteristics that are desirable in both CAD and analysis. However, because each mesh element is a parametric unit square, these splines lack additional geometric flexibility and fidelity available with non-uniform parameterizations. In this work, we extend the framework of Almost C1 splines to account for non-uniform behavior and sharp features that may be present in CAD models. These non-uniform parametric domains can be informed by parameterizations typical in spline reconstruction algorithms. We show that these splines again boast optimal convergence rates while also having potential for higher geometric fidelity than their uniform counterparts.