A continuation method is described for extending the applicability of quasilinearization to numerically unstable two-point boundary-value problems. Since quasilinearization is a realization of Newton's method, one might expect difficulties in finding satisfactory initial trialpoints, which actually are functions over the specified interval that satisfy the boundary conditions. A practical technique for generating suitable initial profiles for quasilinearization is described. Numerical experience with these techniques is reported for two numerically unstable problems.