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In this study, we numerically explore methods of coupling sensitivity analysis to the reduced model in order to increase the accuracy of a proper orthogonal decomposition (POD) basis across a wider range of parameters. Various techniques based on polynomial interpolation and basis alteration are compared. These techniques are performed on a 1-dimensional reaction-diffusion equation and 2-dimensional incompressible Navier-Stokes equations solved using the finite element method (FEM) as the full scale model. The expanded model formed by expanding the POD basis with the orthonormalized basis sensitivity vectors achieves the best mixture of accuracy and computational efficiency among the methods compared.