Continuously and widely tunable lasers, actively stabilized on a frequency reference, are broadly employed in atomic, molecular, and optical (AMO) physics. The frequency-stabilized optical frequency comb (OFC) provides a novel optical frequency reference, with a broadband spectrum that meets the requirement of laser frequency stabilization. Therefore, we demonstrate a frequency-stabilized and precisely tunable laser system based on it. In this scheme, the laser frequency locked to the OFC is driven to jump over the ambiguity zones, which blocks the wide tuning of the locked laser, and tuned until the mode hopping happens with the always-activated feedback loop. Meanwhile, we compensate the gap of the frequency jump with a synchronized acoustic optical modulator to ensure the continuity. This scheme is applied to an external cavity diode laser (ECDL), and we achieve tuning at a rate of about 7 GHz/s, with some readily available commercial electronics. Furthermore, we tune the frequency-stabilized laser only with the feedback of diode current, and its average tuning speed can exceed 100 GHz/s. Due to the resource-efficient configuration and the simplicity of completion, this scheme can be referenced and can find wide applications in AMO experiments.