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Cavity optomechanical system can exhibit higher-order sideband comb effect when it is driven by a control field $ω_{c}$ and a probe field $ω_{p}$, and works in the non-perturbative regime, as was shown in a previous work [Xiong et al., Opt. Lett. 38, 353 (2013)]. The repetition frequency of such a comb is equal to the mechanical frequency $ω_{b}$ and is untunable, which limits the precision of the comb. Here we address this problem by driving the system with an additional strong probe field $ω_{f}$, and the detuning between $ω_{f}$ and $ω_{c}$ is equal to $ω_{b}/n$ (here $n$ is an integer), i.e., this detuning is a fraction of the mechanical frequency. In this case, we obtain some interesting results. We find that not only the integer-order (higher-order) sidebands, but also the fraction-order sidebands, and the sum and difference sidebands between the integer- and fraction-order sidebands, will appear in the output spectrum. The generated nonlinear sidebands constitute an optomechanically induced sideband comb (OMISC). The frequency range and the repetition frequency of the OMISC are proportional to the sideband cutoff-order number and the sideband interval, respectively. We show that we can extend the frequency range of the OMISC by increasing the intensity of the probe field $ω_{p}$. More importantly, we can decrease the repetition frequency, and consequently, improve the precision of the OMISC by increasing $n$ and the intensity of the probe field $ω_{f}$.