Monitoring of changing samples by Cavity Ring-Down Spectroscopy (CRDS) is possible using fast frequency scans of the laser and/or the cavity resonance. Mode-matched cavity excitation improves performance of fast CRDS but data-points result separated by the cavity Free Spectral Range (FSR): low pressure samples demand long cavities. We demonstrate fast CRDS with off-axis injection of a "re-entrant" resonator yielding FSR/N data-points separation. Our N = 4 short-cavity setup is found to perform well compared with other fast-CRDS implementations. Interestingly, the intrinsic chirped ringing affecting ring-down signals in mode-matched fast-CRDS disappear with off-axis injection. This is due to a fine splitting of the re-entrant-cavity degenerate groups of modes by astigmatism.