By far the most popular method of resolving redundant systems involves application of the 2-norm-optimizing Moore-Penrose pseudo-inverse. Pseudo-inverse resolution resolves systems uniquely, continuously, and most notably with simple implementation for general systems. However, resolution using 2-norm fails to exploit general systems' full realizable output space when input bounds are present. The Cascaded Generalized Inverse (CGI) was introduced to address this problem by iteratively reallocating oversaturated input resolutions to other, under-utilized inputs. In this paper, it is demonstrated that although successful in extending the resolution range of 2-norm, CGI still fails to capture the full benefit of a system's realizable output space. To remedy this, an Extended CGI (eCGI) is introduced to allow a larger space of successful resolution through selective desaturation of saturated inputs through CGI. eCGI is demonstrated in resolving kinematic redundancy and was shown in a numerical example to increase the maximum controllable end-effector acceleration over what is possible with CGI to the full physically achievable space.