We present the results of a theoretical study on dynamic Stark-induced coherent pi-electron rotations in a chiral aromatic ring molecule. This is an extension of our previous papers, which have been published in Mineo, H. et al. [Phys. Chem. Chem. Phys. 2016, 18, 26786-26795] and Mineo, H. et al. [J. Phys. Chem. Lett. 2018, 9, 5521-5526]. In those papers, the time-dependent Schrodinger equation was solved under a restricted condition in which a degenerate excited state should be formed at the center of the two relevant excited states by dynamic Stark effects. The dynamic Stark-induced degenerate state (DSIDS) is essential to create unidirectional pi-electron rotations. In the present theoretical treatment, the above restriction is relaxed and the DSIDS is set to be at any energy position between the two excited states. This indicates a wide applicability of the dynamic Stark effects to coherent control of photophysical properties in aromatic molecules, such as coherent ring currents and current-induced magnetic fluxes of low-symmetric aromatic molecules. Analytical expressions for the coherent pi-electron angular momentum are derived within a three-electronic-state model by using the Laplace transform method. The validity of the developed theoretical procedure is demonstrated by carrying out simulations of the coherent angular momentum of L-phenylalanine. Effects of varying the DSIDS on the time-dependent coherent angular momentum and the populations in the three electronic states are examined, and the results are analyzed using approximate expressions for the time-dependent coherent angular momentum and the populations. Modulations in the time-dependent coherent angular momentum appear when the DSIDS is set at an energy position between the two excited states, while there are no beating modulations when the DSIDS is set at the center position. Such differences originate from whether interferences between the two dressed states take place or not.