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Rates of above-threshold detachment (ATD) of H− (s-electron) and F− (p-electron) ions, calculated using the Sturmian-Floquet quasienergy approach, are investigated as a function of intensity for the case that the laser field intensity passes across ponderomotively shifted detachment thresholds. Pronounced enhancements of both detached electron angular distributions and fixed-angle ATD spectra in the low- and mid-energy plateau region are found at near-threshold laser intensities. These enhancements are sensitive to the initial state symmetry (i.e., the s or p symmetry) of the valence electron and occur in a narrow angular interval that is far from the polarization axis at electron energies near the onset of the plateau and moves toward the polarization axis with increasing electron energy. Although enhancements in detached electron energy distributions have not been observed for energies near the cutoff of the ATD plateau, the corresponding electron angular distributions are found to change shape dramatically as a function of intensity near the closing of ponderomotively shifted detachment thresholds. A comparison of our quantum-mechanically calculated angular distributions and those predicted classically in the high-energy region of the ATD plateau is presented and discussed.