The energy spectrum of Galactic cosmic-ray (GCR) ions at Earth varies with solar activity as these ions cross the heliosphere. Thus, this “solar modulation” of GCRs provides remote sensing of heliospheric conditions throughout the ∼11 yr sunspot cycle and ∼22 yr solar magnetic cycle. A neutron monitor (NM) is a stable ground-based detector that measures cosmic-ray rate variations above a geomagnetic or atmospheric cutoff rigidity with high precision (∼0.1%) over such timescales. Furthermore, we developed electronics and analysis techniques to indicate variations in the cosmic-ray spectral index using neutron time-delay data from a single station. Here we study solar modulation using neutron time-delay histograms from two high-altitude NM stations: (1) the Princess Sirindhorn Neutron Monitor at Doi Inthanon, Thailand, with the world’s highest vertical geomagnetic cutoff rigidity, 16.7 GV, from 2007 December to 2018 April; and (2) the South Pole NM, with an atmosphere-limited cutoff of ∼1 GV, from 2013 December to 2018 April. From these histograms, we extract the leader fraction L, i.e., inverse neutron multiplicity, as a proxy of a GCR spectral index above the cutoff. After correction for pressure and precipitable water vapor variations, we find that L roughly correlates with the count rate but also exhibits hysteresis, implying a change in spectral shape after a solar magnetic polarity reversal. Spectral variations due to Forbush decreases, 27 day variations, and a ground-level enhancement are also indicated. These methods enhance the high-precision GCR spectral information from the worldwide NM network and extend it to higher rigidity.