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We report new observations of the cool diffuse gas around 29, $2.3<z<6.3$galaxies, using deep JWST/NIRCam slitless grism spectroscopy around thesightline to the quasar J0100+2802. The galaxies span a stellar mass range of$7.1 \leq \log M_{*}/M_{sun} \leq 10.7$, and star-formation rates of $-0.1 <\log \; SFR/M_{sun}yr^{-1} \; <2.3$. We find galaxies for seven MgII absorptionsystems within 300 kpc of the quasar sightline. The MgII radial absorptionprofile falls off sharply with radii, with most of the absorption extending outto 2-3$R_{200}$ of the host galaxies. Six out of seven MgII absorption systemsare detected around galaxies with $\log M_{*}/M_{sun} >$9. MgII absorptionkinematics are shifted from the systemic redshift of host galaxies with amedian absolute velocity of 135 km/s and standard deviation of 85 km/s. Thehigh kinematic offset and large radial separation ($R> 1.3 R_{200}$), suggestthat five out of the seven MgII absorption systems are gravitationally notbound to the galaxies. In contrast, most cool circumgalactic media at $z<1$ aregravitationally bound. The high incidence of unbound MgII gas in this worksuggests that towards the end of reionization, galaxy halos are in a state ofremarkable disequilibrium, and are highly efficient in enriching theintergalactic medium. Two strongest MgII absorption systems are detected at$z\sim$ 4.22 and 4.5, the former associated with a merging galaxy system andthe latter associated with three kinematically close galaxies. Both thesegalaxies reside in local galaxy over-densities, indicating the presence of coolMgII absorption in two "proto-groups" at $z>4$.

EIGER IV: The cool 10$^4$K circumgalactic environment of high-$z$ galaxies reveals remarkably efficient IGM enrichment