Long-term precession represents the secular motion of the Earth's axis for a long time interval. In 2015, we calculated the Earth's long-term precession in a relativistic framework. However, our previous works involving the ecliptic in the Geocentric Celestial Reference System are deficient, because the natural definition of this ecliptic is still ambiguous. To obtain a long-term precession model in accordance with general relativity, improvements are made including the following: all calculations are no longer carried out in the reference related to any ecliptic; a new hybrid integrator is designed and used for this precession model; ecliptic-independent precession parameters are calculated and provided. A detailed comparison analysis is performed to estimate the importance of various relativistic influences on the precession parameters. Valid within the interval of ± 1 Myr around J2000.0, a consistent post-Newtonian long-term precession model for the Earth has been achieved and is presented here.