We study the resummation of large logarithmic perturbative corrections to the partonic cross sections relevant for the process pp → hX at high transverse momentum of the hadron h, when the initial protons are longitudinally polarized. We perform the resummation to next-to-leading logarithmic accuracy. We present numerical results for center-of-mass energies √ S = 19.4 GeV, relevant for comparisons to data from the Fermilab E704 experiment, and √ S = 62.4 GeV, where preliminary data from RHIC have recently become available. We find significant enhancements of the spindependent cross sections, but a decrease of the double-spin asymmetry for the process. This effect is less pronounced at the higher energy.