Density-functional exchange-correlation approximations depending on spin densities and their gradients have proven remarkably accurate in recent thermochemical tests [e.g., A. D. Becke, J. Chem. Phys. 107, 8554 (1997)]. With the inherent limitations of first-order gradient corrections now in sight, however, we investigate here a class of inhomogeneity corrections based on a new second-order gradient parameter. The new parameter is logically motivated by previous work on Taylor expanded exchange hole densities, and generates exchange-correlation functionals more accurate than those containing first-order gradients only.