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We use optical-optical double-resonance spectroscopy with a continuous wave (CW) pump and a cavity-enhanced frequency comb probe to measure the energy levels of methane in the upper part of the triacontad polyad (P6) with higher rotational quantum numbers than previously assigned. A high-power CW optical parametric oscillator, tunable around 3000 cm-1, is consecutively locked to the P(7, A2), Q(7, A2), R(7, A2), and Q(6, F2) transitions in the ν3 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 3ν3 ← ν3 spectral range with uncertainties down to 300 kHz (1 × 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 with 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 2ν3 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.