This study entailed the successful deployment of a novel neutron interferometer that utilizes multilayer mirrors. The apparatus facilitates a precise evaluation of the wavelength dependence of interference fringes utilizing a pulsed neutron source. Our interferometer achieved an impressive precision of 0.02 rad within a 20-min recording time. Compared to systems using silicon crystals, the measurement sensitivity was maintained even when using a simplified disturbance suppressor. By segregating beam paths entirely, we achieved successful measurements of neutron-nuclear scattering lengths across various samples. The values measured for Si, Al, and Ti were in agreement with those found in the literature, while V showed a disparity of 45%. This discrepancy may be attributable to impurities encountered in previous investigations. The accuracy of measurements can be enhanced further by mitigating systematic uncertainties that are associated with neutron wavelength, sample impurity, and thickness. This novel neutron interferometer enables us to measure fundamental parameters, such as the neutron-nuclear scattering length of materials, with a precision that surpasses that of conventional interferometers.