High-redshift blazars ($z\geq2.5$) are the most powerful class of persistent $\gamma$-ray sources in the Universe. These objects possess the highest jet powers and luminosities and have black hole masses often in excess of $10^9$ solar masses. In addition, high-$z$ blazars are important cosmological probes and serve as test objects for blazar evolution models. Due to their large distance, their high-energy emission is typically scattered to energies below the GeV range, which makes them difficult to study with Fermi/LAT and only the very brightest objects are detectable. Hence, only a small number of high-$z$ blazars could be detected with Fermi/LAT so far. In this work, we present a strategy to increase the detection statistics at redshift $z\geq2.5$ via a search for flaring events in high-$z$ $\gamma$-ray blazars whose long-term averaged flux is just below the sensitivity limit of Fermi/LAT. Seven previously GeV undetected high-$z$ blazars have been identified with a test statistic of TS $\geq\,25$ from their bright monthly outburst periods, while more detections are expected in the future.