The emission of very-high-energy (VHE) photons ($E\ \gt\ 100\ \mathrm{GeV}$) in blazars is closely connected to the production of ultra-relativistic particles and the role of these γ-ray sources as cosmic particle accelerators. This work focuses on a selection of 22 γ-ray objects from the 2BIGB catalogue of high-synchrotron-peaked sources, which are classified as blazar candidates of uncertain type in the 4FGL-DR2 catalogue. We study these sources by means of a re-analysis of the first 10 yr of γ-ray data taken with the Fermi-Large Area Telescope, including the attenuation by the extragalactic background light. Their broad-band spectral energy distributions are also evaluated, using multiwavelength archival data in the radio, optical, and X-ray bands, in terms of one-zone synchrotron-self-Compton models, adding an external Compton component when needed. Out of this analysis, we identify 17 new extreme high-synchrotron-peaked (EHSP) candidates and compare their physical parameters with those of prototypical EHSP blazars. Finally, the resulting models are used to assess their detectability by the present and future generation of ground-based Imaging Atmospheric Cherenkov Telescopes. We find two VHE candidates within the reach of the current and next generation of Cherenkov telescopes: J0847.0-2336 and J1714.0-2029.