We use sub-Doppler optical-optical double-resonance (OODR) spectroscopy with a 3.3 µm single-frequency pump and a cavity-enhanced 1.65 µm comb probe to measure 33 ladder-type (3ν3 ← ν3) and 8 V-type (2ν3) transitions in the 5880–6090 cm-1 range of methane, reaching states with rovibrational E symmetry in the region of the P 6 and P 4 polyads, respectively. We assign the ladder-type transitions using new Hamiltonian predictions and the ExoMol line list, and the V-type transitions using the new Hamiltonian, ExoMol, HITRAN2020, and the WKLMC line lists. While 7 of the states in the 3ν3 range have been previously observed either in earlier OODR work (without cavity enhancement) with 1.5 MHz accuracy or in FTIR measurements of cold bands with 150 MHz resolution, the states reported here have uncertainties down to 150 kHz (5 × 10–6 cm-1). The E-symmetry states exhibit first-order Stark splitting, which will be reported in our future work.