The solar modulation of Galactic cosmic rays (GCRs) provides remote sensing of heliospheric conditions throughout the $\sim$11-yr sunspot cycle and $\sim$22-yr solar magnetic cycle. Neutron monitors (NMs) can measure cosmic ray rates above a cutoff rigidity with high precision ($\sim$0.1\%) over such time scales. To avoid systematic uncertainties in comparing NM count rates from different stations, here we study solar modulation of the GCR spectrum by independently using neutron time-delay histograms to determine the leader fraction $L$, i.e., the inverse neutron multiplicity, as a proxy of the GCR spectral index above the cutoff. Data were collected using specialized electronics at two high-altitude NM stations: 1) the Princess Sirindhorn Neutron Monitor (PSNM) at Doi Inthanon, Thailand, with the world's highest vertical geomagnetic cutoff rigidity, 16.8 GV, from 2007 December to 2018 September, and 2) the South Pole Neutron Monitor with an atmosphere-limited cutoff of $\sim$1 GV, from 2013 December to 2018 September. After correcting for pressure and (in Thailand) precipitable water vapor, we find that $L$ is roughly correlated with the count rate, but also exhibits hysteresis that indicates a change in spectral shape after the change in solar magnetic polarity. These methods enhance the high-precision GCR spectral information from the worldwide NM network and extend it to higher rigidity.