We propose a different way to satisfy both gradient expansion limiting behavior and the Lieb-Oxford bound in a generalized gradient approximation exchange functional by extension of the Perdew-Burke-Ernzerhof (PBE) form. Motivation includes early and recent exploration of modified values for the gradient expansion coefficient in the PBE exchange-correlation functional (cf. the PBEsol functional) and earlier experience with a numerical cutoff for large-s (s proportional to absolute value(vector differential n)/n(4/3)) in a version of the deMon molecular code. For either the original PBE or the PBEsol choice of the gradient coefficient, we find improved performance from using an s-dependent (spatially varying) satisfaction of the Lieb-Oxford bound which quenches to uniform electron gas behavior at large s. The mean absolute deviations (MADs) in atomization energies for a widely used test set of 20 small molecules are reduced by about 22% relative to PBE and PBEsol. For these small molecules, the bond length MADs are essentially unchanged.