The improved primal simplex (IPS) was recently developed by Elhalaloui et al. to take advantage of degeneracy when solving linear programs with the primal simplex. It implements a dynamic constraint reduction based on compatible variables whose identification may be computationally costly. In this article, we first show how the positive edge criterion of Raymond et al. can be included in IPS for a fast identification of the compatible variables. Our algorithm then proceeds through a series of reduction and augmentation phases until optimality is reached. In a reduction phase, we identify compatible variables and focus on them to make quick progress toward optimality. During an augmentation phase, we compute a greatest normalized improving direction and select a subset of variables that should be considered in the reduced problem. This new algorithm is tested over Mittelmann's benchmark for linear programming and on instances arising from industrial applications. The results show that the new algorithm outperforms CPLEX on every highly degenerate instance in which a sufficient number of nonbasic variables are compatible. In contrast, IPS has difficulties on the eleven largest Mittelmann instances.