The orthogonal two-step annealing process is an effective strategy to linearize the response of magnetic tunnel junctions for magnetic field sensors. However, the response after the orthogonal annealing is inevitably modulated by the Neel effect from the reference layer, which results in an unexpected shift of the linear interval and a disappointing sensitivity deterioration in the weak field. Here, a non-orthogonal two-step annealing method is proposed to suppress the shift by compensating for the Neel coupling field. Experimental results show that the curve shift of junctions annealed in the non-orthogonal direction of 120° is 47.6% lower than that in the orthogonal one, with a significant sensitivity promotion in the weak field and little hysteresis increment. A simple energy minimization model is introduced to explain the results. Based on the model, the suppression of the curve shift is fulfilled with the effective field compensation for the Neel coupling field, modulated by the non-orthogonal annealing. Finally, Wheatstone bridge devices are constructed, and the bridge with non-orthogonally annealed junctions is found to have an increased sensitivity of 46.8% in the major loop along the sensing axis. Additionally, the non-orthogonal annealing method is also effective in suppressing the cross sensitivity, which is important for further application to three-axis magnetic sensors.