A search for a narrow resonance with a mass between 350 and 700 GeV, and decaying into a pair of jets, is performed using proton-proton collision events containing at least three jets. The data sample corresponds to an integrated luminosity of $18.3 ~\mathrm{fb}^{-1}$ recorded at $\sqrt{s}=13~\mathrm{TeV}$ with the CMS detector. Data are collected with a technique known as "data scouting", in which the events are reconstructed, selected, and recorded at a high rate in a compact form by the high-level trigger. The three-jet final state provides sensitivity to lower resonance masses than in previous searches using the data scouting technique. The spectrum of the dijet invariant mass, calculated from the two jets with the largest transverse momenta in the event, is used to search for a resonance. No significant excess over a smoothly falling background is found. Limits at 95% confidence level are set on the production cross section of a narrow dijet resonance. The corresponding upper limits on the coupling of a narrow vector resonance interacting only with quarks are between 0.10 and 0.15, depending on the resonance mass.These results represent the most stringent upper limits in the mass range between 350 and 450 GeV obtained with a search that is sensitive to the decay of the resonance into light-flavor quarks.