High-redshift blazars ($z\geq2.5$) are one of the most powerful classes of $\gamma$-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-redshift blazars are important cosmological probes and serve as test objects for blazar evolution models. Due to their large distance, their high-energy emission peaks are often downshifted 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-redshift 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-redshift $\gamma$-ray blazars whose long-term flux is just below the sensitivity limit of Fermi/LAT. Seven previously GeV undetected high-redshift blazars have been identified from their bright monthly outburst periods, while more detections are expected in the future.