We present a new framework for extending Chapman-Enskog theory beyond the hard-sphere fluid model. Rather than relying on effective hard sphere diameters, the approach makes use of an exchange function which can be related to the thermodynamic properties of the system. We show that two existing extensions, including modified Enskog theory (MET), fit into this new framework. Based on our approach, we propose an alternative to MET that takes into account the potential interaction energy associated with the inter-particle interactions in the fluid. The proposed expression is applied to predicting the shear viscosity of several different simulated fluid models across a wide set of densities 0.05 <= rho* <= 0.8 and temperatures 1.5 <= T* <= 4.0 in Lennard-Jones units. The fluid models considered include both the Weeks-Chandler-Anderson (WCA) fluid and the Lennard-Jones (LJ) fluid. At low and intermediate density, here taken to be rho* <= 0.3, we report mean relative prediction errors between 2% and 4% for both these. Across all densities considered, the largest mean relative errors reported are 4.4% and 8.1% for the WCA fluid and LJ fluid respectively. We also investigate other interaction models, including a diatomic molecular model, in order to better understand the limitations of our approach.