The rapid and accurate initial alignment of a strapdown inertial navigation system (SINS) aided by a Doppler velocity logger (DVL) still poses a significant challenge in unmanned underwater vehicle navigation. In this study, a novel adaptive nonlinear filtering alignment scheme is developed for a SINS with a large misalignment angle. In previous studies, the frame-inconsistency problem of the nonlinear SINS error model was neglected. Thus, a modified nonlinear error model of a SINS is derived here rigorously as the process model of the nonlinear Kalman filter. In addition, the noise amplification of applying Sage-Husa adaptive Kalman filter (SHAKF) in alignment is analyzed, and a novel arctangent fading memorial factor is introduced to dynamically adjust the weight of current measurements. Simulations and field experiments indicate that the proposed algorithm is efficient in underwater DVL/SINS in-motion alignment. The alignment accuracy and convergence rate of the modified model are significantly improved compared to those of the traditional model. The arctangent fading memorial SHAKF can reliably estimate the measurement noise and improve the performance of alignment.