In the state estimation of a nonlinear system, the second-order filter is known to achieve better precision than the first-order filter [extended Kalman filter (EKF)] at the price of complex computation. If the measurement equation is linear in a transformed state variable, the complex measurement update equations of the second-order filter become as simple as the EKF case. Further, if the vector fields carrying the noise are constant, the high-order components in the variance propagation equation disappear. This suggests that if we make the measurement equation linear and make some vector fields constant through a coordinate transformation, we can simplify the second-order filter significantly while taking advantage of high precision. Finally, with an example of a falling body, we demonstrate through a Monte Carlo analysis the usefulness of the proposed method.