  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 fields 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-size or medium-size telescopes to the
network at diverse geographic locations taking into account the positions of
CTA and the LIGO-Virgo-KAGRA network. (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
for directions above and 'below' the United States in which the present network
of gravitational-wave detectors is more sensitive.
