In this paper, we show that minimax rational approximations can be enhanced by introducing a controlling parameter on the denominator of the rational function. This is implemented by adding a small set of linear constraints to the underlying optimization problem. The modification integrates naturally into approximation models formulated as linear programming problems. We demonstrate our approach through several numerical examples, including a multivariate rational approximation of the Korteweg–de Vries (KdV) equation from fluid dynamics.