  Hyperfine-induced electric dipole contributions may significantly increase
probabilities of otherwise very weak electric octupole and magnetic quadrupole
atomic clock transitions (e.g. transitions between $s$ and $f$ electron
orbitals). These transitions can be used for exceptionally accurate atomic
clocks, quantum information processing and search for dark matter. They are
very sensitive to new physics beyond the Standard Model, such as temporal
variation of the fine structure constant, the Lorentz invariance and Einstein
equivalence principle violation. We formulate conditions under which the
hyperfine-induced electric dipole contribution dominates. Due to the hyperfine
quenching the electric octupole clock transition in $^{173}$Yb$^+$ is two
orders of magnitude stronger than that in currently used $^{171}$Yb$^+$. Some
enhancement is found in $^{143}$Nd$^{13+}$, $^{149}$Pm$^{14+}$,
$^{147}$Sm$^{14+}$, and $^{147}$Sm$^{15+}$ ions.
