The Photonic hook (PH) is an intricately curved photonic nanojet (PNJ) or a highly intense electromagnetic beam featuring a subwavelength waist, whose principal hallmark lies in its capacity to bend light at the nanoscale. According to existing literature, the origin of PH can be attributed to symmetry breaking, whereas symmetrical microstructures predominantly contribute to PNJ formation. This study presents the novel revelation of PH emergence from an isolated eccentric core-shell dielectric microcylinder, achieved through the illumination of a paraxial Gaussian beam (PGB). The eccentrically structured core-shell microscale geometry introduces an additional degree of freedom, influencing PH formation and directly shaping its characteristic parameters. Much like PNJ, the propagation of PH depends on different parameters such as core and shell refractive indices of the micro-structures, microstructure geometry, incident light type, and direction of propagation. A fascinating outcome from our numerical simulations is the switchable occurrence of PNJ and PH from an eccentric core-shell microcylinder by a simple adjustment of eccentricity, either parallel or perpendicular to the PGB's propagation direction. This computational investigation emphasizes the impact of eccentricity and the incident wave's beam waist, maintaining a consistent refractive index contrast between the core and shell. The outcomes are interpreted in terms of key parameters governing PH generation characteristics, encompassing FWHM, maximum electric field enhancement, and focal plane. Notably, we have observed the coexistence of whispering gallery modes (WGM) and PH within this system and these modes exhibit high sensitivity to the excitation wavelength. The potential applications of PH are believed to be far-reaching, including areas like optical trapping, sensing, and functioning as a versatile focusing element. This study not only contributes to our fundamental understanding of PH but also illuminates its potential as a robust tool across diverse optical applications.