Traditional spacecraft attitude estimation algorithms normally require the assumption that the true quaternion approximates to the estimated value. Otherwise, it may lead to a large truncation error. However, this assumption is difficult to be always satisfied during practical applications. To alleviate it, this article proposes a new scheme to reconstruct the error covariance matrix to minimize this truncation error. Specifically, the recursive formulas of measurement updates used in traditional algorithms are recalculated with the quaternion derived from the attitude update, which can reduce the influence caused by the initial conditional errors and inaccurate attitude estimations. This new scheme is then applied to the quaternion attitude estimators. It is shown that the error covariances of the proposed algorithms are theoretically bounded. Simulations for a low-earth-orbit spacecraft state estimation are performed. Through updating the estimated quaternion and then reconstructing the error covariance, the comparison results show that the proposed algorithms thus reduce the introduction of truncation error and thus achieve better performance than traditional algorithms in terms of robustness and convergence rate.