Quadrupole moments of six high-spin isomers in the At isotopes have been measured with the level-mixing-spectroscopy method: At-208(16-), At-209(29/2+), At-210(19+, 15-). At-211 (39/2-, 29/2+). The results show that level mixing spectroscopy is a promising technique to determine quadrupole moments of isomers that are difficult to measure by other in-beam hyperfine interaction methods. A large increase of the quadrupole moment is observed if neutrons are excited across or removed from the N = 126 shell closure. This behavior is explained in terms of an enhanced core softness for fewer core neutrons; the aligned valence particles, moving in equatorial orbits, then easily polarize the core towards oblate deformation.