Context. Magnetic fields have only recently been included in theoretical simulations of high-mass star formation. The simulations show that magnetic fields play an important role in the formation and dynamics of molecular outflows. Masers, in particular 6.7-GHz CH 3 OH masers, are the best probes of the magnetic field morphologies around massive young stellar objects on the smallest scales of 10-100 AU. Aims. Providing new observational measurements of the morphology of magnetic fields around massive young stellar objects by using 6.7-GHz CH 3 OH maser emission is very important for setting constraints on the numerical simulations of massive star formation. Methods. This paper focuses on 4 massive young stellar objects, IRAS 06058+2138-NIRS 1, IRAS 22272+6358A, S255-IR, and S231, which complement our previous 2012 sample (the first EVN group). From all these sources, molecular outflows have been detected in the past. Seven of the European VLBI Network antennas were used to measure the linear polarization and Zeeman-splitting of the 6.7-GHz CH 3 OH masers in the star-forming regions in this second EVN group. Results. We detected a total of 128 CH 3 OH masing cloudlets. Fractional linear polarization (0.8%-11.3%) was detected towards 18% of the CH 3 OH masers in our sample. The linear polarization vectors are well ordered in all the massive young stellar objects. We measured significant Zeeman-splitting in IRAS 06058+2138-NIRS 1 (ΔV Z = 3.8 ± 0.6 m s −1 ) and S255-IR (ΔV Z = 3.2 ± 0.7 m s −1 ). Conclusions. By considering the 20 massive young stellar objects towards which the morphology of magnetic fields was determined by observing 6.7-GHz CH 3 OH masers in both hemispheres, we find no evident correlation between the linear distributions of CH 3 OH masers and the outflows or the linear polarization vectors. On the other hand, we present first statistical evidence that the magnetic field (on scales 10-100 AU) is primarily oriented along the large-scale outflow direction. Moreover, we empirically find that the linear polarization fraction of unsaturated CH 3 OH masers is P l < 4.5%.