This paper proposes a new method for gear tooth contact analysis, which concentrates on solving two considerable disadvantages of the generalized algorithm: 1) numerical instability due to adopting a system of five nonlinear equations to search for each contact point; 2) complexity of the computation process caused by differences in the mathematical models for the tooth surface contact and edge contact. To address the first issue, the instantaneous conjugate contact curve, and its modified curve, were defined and utilized to effectively separate the transmission error; thus, the contact point and instantaneous contact curve can be effortlessly located and the number of required nonlinear equations is reduced from 5 to 2. The second disadvantage was addressed using a universal mathematical model for both the tooth surface contact and edge contact. Finally, the proposed algorithm was used to simulate the meshing of a face-milled spiral bevel gear set and similar results to using the Gleason software were attained. Real contact patterns were verified with the gear-rolling test, and comparing these results to our simulations, the effectiveness of the proposed methodology was demonstrated.