Frequency-modulated continuous-wave (FMCW) can be acquired by using a distributed feedback semiconductor laser (DFB-SL) operating at period-one (P1) oscillation under an optical injection modulated by a Mach-Zehnder modulator (MZM). In this work, through introducing another MZM to establish cascade-modulated optical injection, an improved photonic scheme for generating high-quality FMCW is proposed and experimentally demonstrated. The experimental results indicate that, under appropriate injection parameters, the central frequency of the generated FMCW is widely tunable, and the bandwidth is larger than that obtained under a single MZM modulation. Further introducing optical feedback for suppressing the phase noise, the frequency comb contrast of the generated FMCW can be improved obviously.