Observations suggested that the occurrence rate of hot Jupiters (HJs) in open clusters is largely consistent with the field ($\sim 1\%$) but in the binary-rich cluster M67, the rate is $\sim 5\%$. How does the cluster environment boost HJ formation via the high-eccentricity tidal migration initiated by the extreme-amplitude von Zeipel-Lidov-Kozai (XZKL) mechanism forced by a companion star? Our analytical treatment shows that the cluster’s collective gravitational potential alters the companion’s orbit slowly, which may render the star-planet-companion configuration XZKL-favourable. We have also performed direct Gyr N-body simulations of the star cluster evolution and XZKL of planets’ orbit around member stars. We find that an initially-single star may acquire a companion star via stellar scattering and the companion may enable XZKL in the planets’ orbit. Planets around an initially-binary star may also be XZKL-activated by the companion. In both scenarios, the companion’s orbit has likely been significantly changed by stellar scattering and the cluster potential before XZKL occurs. Across different cluster models, 0.8%-3% of the planets orbiting initially-single stars have experienced XZKL while the fraction is 2%-26% for initially-binary stars. Around a star that is binary at 1 Gyr, 13%-32% of its planets have undergone XZKL, and combined with single stars, the overall XZKL fraction is 3%-21%, most affected by the cluster binarity. If 10% of the stars in M67 host a giant planet, our model predicts an HJ occurrence rate of $\sim 1\%$. We suggest that HJ surveys target old, high-binarity, not-too-dense open clusters and prioritise wide binaries to maximise HJ yield.