  We use optical-optical double-resonance (OODR) spectroscopy with a continuous
wave (CW) pump and a cavity-enhanced frequency comb probe to measure high
rotational energy levels of methane in the upper part of the triacontad polyad
(P6). A high-power CW optical parametric oscillator, tunable around 3000
cm$^{-1}$, is consecutively locked to the P(7, A$_2$), Q(7, A$_2$), R(7,
A$_2$), and Q(6, F$_2$) transitions in the ${\nu}$$_3$ band, and a comb
covering the 5800-6100 cm$^{-1}$ range probes sub-Doppler ladder-type
transitions from the pumped levels with J' = 6 to 8, respectively. We report
118 probe transitions in the 3${\nu}$$_3$ $\leftarrow$ ${\nu}$$_3$ spectral
range with uncertainties down to 300 kHz (1 x 10$^{-5}$ cm$^{-1}$), reaching 84
unique final states in the 9070-9370 cm$^{-1}$ range with rotational quantum
numbers J between 5 and 9. We assign these states using combination differences
and by comparison to theoretical predictions from a new ab initio-based
effective Hamiltonian and dipole moment operator. This is the first
line-by-line experimental verification of theoretical predictions for these
hot-band transitions, and we find a better agreement of transition wavenumbers
with the new calculations compared to the TheoReTS/HITEMP and ExoMol databases.
We also compare the relative intensities and find an overall good agreement
with all three sets of predictions. Finally, we report the wavenumbers of 27
transitions in the 2${\nu}$$_3$ spectral range, observed as V-type transitions
from the ground state, and compare them to the new Hamiltonian, HITRAN2020,
ExoMol and the WKMLC line lists.
