Comb Coherence Transfer (CCT) uses a feed-forward frequency correction to transfer the optical phase of a frequency comb to the beam of a free-running diode laser. This allows the amplification of a selected comb tooth by 50 dB while adding agile and accurate frequency tuning. In the present work, SI-traceable frequency calibration and comb tooth narrowing down to 20 kHz is additionally provided by comb frequency locking to an ultrastable optical frequency reference distributed from Paris to Grenoble through the RENATER optical fiber network [Lisdat et al., Nat. Commun., 2016, 7, 12443]. We apply this CCT broadly tunable source for saturated cavity ring-down spectroscopy of ro-vibrational R0 to R10 multiplets in the 2ν3 band of 12CH4 (from 6015 to 6115 cm-1). Indeed, efficient cavity injection with large intra-cavity power build-up induces saturation of the ro-vibrational transitions at low pressure and Doppler-free Lamb dips are observed with high signal/noise. kHz-accurate transition frequencies are derived improving by three orders of magnitude previous values from spectra in the Doppler regime, which are strongly affected by line blending. While previous saturation spectroscopy investigations addressed specific 2ν3 multiplets (R6 or R9), the CCT approach allowed for a rapid coverage of the entire R0-R10 series. Measured transition frequencies are compared with experimental and theoretical line lists available in the literature.