We applied Raman spectroscopy to investigate the response to electrochemical doping of the second-order D* band in single-walled carbon nanotube (SWNT) bundles. Our study reveals a dramatic increase of the D* band sensitivity to doping upon moving the laser excitation to the red end of the visible spectrum and beyond. Using the double-resonance scattering model, we show that this phenomenon evidences a second Kohn anomaly in metallic SWNTs, located in the K-point-derived region of the Brillouin zone (BZ), which stems from the Kohn anomaly at the K-point of graphene. Our results will be compared to recent doping experiments on graphene with field-effect gating and can be used to investigate the wave-vector dependent electron-phonon coupling in the bulk of the BZ of metallic SWNTs.