Medium- to long-chain alkanes can form upon cooling intermediate phases between isotropic liquid and solid crystalline, called rotator phases, where relative freedom of the molecules to rotate about their long axis is combined with long range translational order. Rotator phases are well documented experimentally but the mechanism of their formation at the molecular level is still not fully explained. In a previous work, we have shown that molecular dynamics simulations can produce rotator phases upon cooling of hexadecane [S. Iliev et al., J. Col. Int. Sci., 2023, 638, 743]. The aim of the current work is to develop a procedure to identify the specific ordered phase obtained in the simulations. The influence of the cooling rate on the freezing process of hexadecane (bulk and surfactant-interfaced to water) is tested as well. Several parameters are combined to quantify the degree of ordering and the type of phase in the studied systems. These are the tilt angle of the molecules with respect to the crystallite plane, the radial distribution function of the centre of mass of the molecules in the crystallite, the percentage of the gauche torsion angles in the molecules, the angle of the second principal axis of each molecule with respect to the x axis of the coordinate system, and estimates from Voronoi analysis. The results show that the systems form a rotator phase, which transitions gradually towards the thermodynamically most stable triclinic crystal, and the transformation progresses to different extent depending on the system. The influence of the cooling rate is related only to the size of the largest crystallite formed, the other parameters of the freezing process remain unaffected. The work also presents a robust procedure for obtaining and identifying different types of ordered phases in alkane-containing systems with thoroughly tested computational protocol and a comprehensive set of structural analyses. Several key characteristics are advanced, compared to previous research [Ryckaert et al., Mol. Phys., 1989, 67, 957; Wentzel et al., J. Chem. Phys. 2011, 134, 224504], namely, a new methodology is proposed to compute the unit cell deformation parameter and azimuthal angle from MD simulation trajectories of the freezing process in alkane-containing systems. The suggested structural analysis, which is independent of the coordinate system, is applicable to any linear-chain system with polycrystalline structure.