A sharp jump in the magnetic response near the first critical twins magnetic field Hic1 was generated under the combined action of a constant biased and a local oscillating damped magnetic fields on the surface of the YBa2Cu3O7-x epitaxial film. The possibility of increasing due to this jump the sensitivity of a standard Hall magnetometer from ~2.5·10-3 Gs/Hz1/2 to ~8·10-7 Gs/Hz1/2 in the frequency range ~1 kHz has been demonstrated. The range of the measured magnetic fields of the magnetometer was extended to the region of weak fields ~(8·10-7-2.5·10-3) Oe, while maintaining the accuracy and linearity of the order of ~0.01%. To increase the sensitivity, linearity and noise sustainability the beginning of the operating point of the magnetometer was shifted to the twins values Hic1. The increased spatial resolution of the magnetometer, determined by the linear dimensions of the YBa2Cu3O7-x film twins, was provided by an increase in the total magnetic field directed perpendicular to the film surfaces and the Hall transducer and reached ~300 nm. Keywords: the twin high temperature superconductor, the density jump of the trapped magnetic flux, biased magnetic field, oscillating damped local magnetic bias field, two-stage Hall magnetometer.