A bandwidth-enhanced linear frequency-modulated (LFM) waveform generation scheme is proposed and demonstrated based on dynamic control of an optically injected semiconductor laser (OISL). The OISL operates at the period-one (P1) oscillation state under proper injection conditions. After photodetection, a tunable microwave signal is obtained with its frequency determined by the optical injection strength and the detuning frequency between the master and slave lasers. For a fixed detuning frequency, an LFM waveform can be generated by introducing an electrical control signal S(t) with a quasi-sawtooth profile to dynamically manipulate the injection strength of the OISL. Then, to overcome the bandwidth limitation by the achievable P1 frequency range under a given detuning frequency, both the injection strength and the detuning frequency are dynamically controlled to achieve a synthesized P1 frequency range, thus generating LFM waveforms with enhanced bandwidths. In our demonstration, LFM waveforms with a synthesized bandwidth of 8 GHz (12-20 GHz) and 24.8 GHz (12.6-37.4 GHz) are generated in the experiment and simulation, respectively.