A three-dimensional (3D) laser cooling method of fast stored ion beams based on a linear coupling mechanism is explored. We extensively study two approaches proposed in previous publications, i.e., the dispersive coupling scheme and the coupling-cavity scheme, confirming how much one can improve the transverse cooling efficiency. A possible design of a coupling cavity is presented. We employ the tracking code SAD and the molecular dynamics code SOLID to carry out reliable numerical experiments where realistic lattice structures of storage rings and particle Coulomb interactions are taken into account. Through systematic simulations, it is demonstrated that resonant coupling remarkably enhances transverse cooling rates for any initial beams, making it feasible to reach an equilibrium temperature far below the current achievable level. We further emphasize the crucial importance of avoiding the Mathieu instability. We also discuss the minimum cooling power required for beam crystallization as well as on an interpretation of past experimental results in the TSR and ASTRID storage rings.