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The recent discovery of TeV emission from gamma-ray bursts (GRBs) by the MAGIC and H.E.S.S. Cherenkov telescopes confirmed that emission from these transients can extend to very high energies. The TeV energy domain reaches the most sensitive band of the Cherenkov Telescope Array (CTA). This newly anticipated, improved sensitivity will enhance the prospects of gravitational-wave follow-up observations by CTA to probe particle acceleration and high-energy emission from binary black hole and neutron star mergers, and stellar core-collapse events. Here we discuss the implications of TeV emission on the most promising strategies of choice for the gravitational-wave follow-up effort for CTA and Cherenkov telescopes more broadly. We find that TeV emission (i) may allow more than an hour of delay between the gravitational-wave event and the start of CTA observations; (ii) enables the use of CTA’s small size telescopes that have the largest field of view. We characterize the number of pointings needed to find a counterpart. (iii) We compute the annual follow-up time requirements and find that prioritization will be needed. (iv) Even a few telescopes could detect sufficiently nearby counterparts, raising the possibility of adding a handful of small-sized or medium-sized telescopes to the network at diverse geographic locations. (v) The continued operation of VERITAS/H.E.S.S./MAGIC would be a useful compliment to CTA’s follow-up capabilities by increasing the sky area that can be rapidly covered, especially in the United States and Australia, in which the present network of gravitational-wave detectors is more sensitive.