The evaluation of a measurement method of linear birefringence with increased sensitivity is presented. The examined method is based on a substantial change of the geometrical phase caused by a small change of an examined medium's birefringence. The measuring setup consists of a linear polarizer, Wollaston compensator, and circular analyzer. The measurement is performed by tracking the phase shift of a fringe pattern. Specific orientation of the elements modifies the setup's response-the sensitivity of the setup can be controlled in a limited measuring range. The present paper concentrates on the factors affecting the setup's sensitivity as well as the accuracy of obtained results. The validity of the proposed approach has been demonstrated by measuring a phase retardance introduced by the liquid crystal retarder. A thousandfold increase in sensitivity has been obtained in the presented experiments, which allows the measurement of retardance introduced by the linear birefringent medium with an accuracy of 0.003° within the limited measuring range.