We report an analysis of the thickness dependence of the liquid-to-hexatic phase transition in a quasi-two-dimensional hard-sphere colloid system as the confining wall separation changes from 1 to 1.6 hard-sphere diameters. In our theoretical evaluation, we study the bifurcation of solutions to the integral equation for the pair correlation function. Our study predicts that at small wall separation the liquid-to-hexatic phase transition is continuous and that it occurs at lower density than the liquid-to-crystal phase transition density, in agreement with the predictions for a strictly two-dimensional system obtained from the Kosterlitz-Thouless-Halperin-Nelson-Young theory. At larger wall separation (larger than about 1.4 hard-sphere diameters), the liquid-to-hexatic phase transition density is predicted to occur at higher density than the liquid-to-crystal phase transition.