The choice of fitting methods for elliptically scattered data obtained with displacement-measuring homodyne quadrature laser interferometers significantly influences the accuracy of the interferometer. This is especially important when the data contain a lot of noise or provide only a segment of the ellipse. The ellipse parameters extracted by the fitting are used either to correct the data or the basic arctangent phase-unwrapping function in order to enhance the accuracy of the measured displacement by reducing the common nonlinearities. We propose the use of linear, ellipse-specific, least-squares fitting that is further bias-corrected using a linear algorithm. This stable fitting method provides a good balance between the accuracy of the fit and the computational efficiency, and never returns corrupt, non-ellipse parameters. It is therefore applicable for an online, uniform fringe subdivision when there is a demand for sub-nanometric resolution. An experimental confirmation of the improvement over traditional fitting methods was carried out with a single-pass, two-detector homodyne quadrature laser interferometer. We were able to operate the interferometer with nanometric accuracy, provided the data draw out at least a quarter-arc of an ellipse.