Complex interfacial physics arising from geometric curvature associated with surface tension as well as phase transformation make it a formidable task to design an accurate, reliable, and yet simple method for direct computation of multiphase flows. Hybrid methods mixing conventional, Volume-of-Fluid, Level Set, Phase Field, and Front Tracking methods have recently become popular in an attempt to overcome the shortcomings of each method alone. We developed the Level Contour Reconstruction Method (LCRM) as part of a hybrid method for treating the complex interface geometry associated with general three-dimensional multi phase flows. The main idea in that work focused on a simple and robust algorithm especially suited for dynamic interfaces in the three-dimensional case by combining characteristics of both Front Tracking and Level Set methods. In this article we describe a modification to the LCRM which introduces a high order interpolation kernel during the course of the interface reconstruction along with a new hybrid surface tension formulation. With this we can essentially eliminate any mass redistribution between regions of differing curvature and reconstruct the interface accurately and smoothly. The improvement with high order reconstruction is also noticeable vis a vis spurious currents which are further decreased by two orders of magnitude over the previous linear reconstruction method. Moreover, there is no disturbance concurrent with reconstruction and the solution fidelity is not influenced by the reconstruction time step. This High Order Level Contour Reconstruction Method retains the simplicity of the original LCRM and avoids complicated interface smoothing procedures.