An issue in the well-known traditional "crisscross" interaction theory frequently used to describe the boundary layer instability development over curved surfaces is that a strong singularity exists, even in the linear regime, which manifests itself in the form of infinite growth of self-excited oscillations for the wave numbers l = O(1) and k --> infinity. This implies that something essential is missing in the formulation, making the Cauchy problem mathematically ill posed, and this, in turn, casts doubts on the validity of calculations made earlier by several researchers using the crisscross interaction theory for investigating laminar-turbulent transition. The derivation of the theory, therefore, needs re-examining. In this article, a key approach is taken; namely, a uniformly valid composite asymptotic expansion procedure is proposed in an effort to suppress the unrealistic amplification of the disturbances at any time in space. As a matter of fact. the triple-deck structure of the disturbance field remains intact as a whole and serves as a basis for the extended asymptotic theory. The backbone of the process inherently involves restoring the longitudinal pressure gradient term accounting for the secondorder approximation in asymptotic expansions for the outermost and innermost sublayers of the conventional crisscross interaction region. The new system as a result has additional terms depending on a small parameter based on the local reference Reynolds number and the curvature of the surface. The new Cauchy model is eventually free from any singularity in the context of the composite approach. The modified linear dispersion relation is obtained and treated both analytically and numerically, and it is verified that the proposed model becomes well posed for a Suitably chosen additional parameter.