We demonstrate an improved concept for nearly bandwidth-limited nonlinear pulse compression down to the few-cycle regime in a fiber chain with alternating sign of dispersion. Whereas the normally dispersive fiber segments generate bandwidth via self-phase modulation, the anomalously dispersive fiber segments recompress the broadened spectral bandwidth by an appropriate amount of group velocity dispersion. Nonlinear pulse compression from 80 fs input pulses to nearly bandwidth-limited 25 fs pulses at 1560 nm was achieved, resulting in a pulse compression factor of 3.2. The use of a specific dispersion-compensating fiber eliminated the impact of higher-order dispersion, such that a high spectral coherence was ensured. We show that nonlinear Schrödinger equation simulations were in good agreement with the experimental results and investigated the transfer of input fluctuations to the output. The concept is transferable to longer input pulse durations, resulting in compression factors of 83 for 10 ps input pulses.