Two recent experiments have demonstrated the phenomenon of dynamical tunneling of cold atoms interacting with standing electromagnetic waves. We show by quantitative calculations that one can achieve a control of the tunneling period over an orders of magnitude range by changing the frequency difference of the waves by about 10% only. In this narrow parameter region, the mechanism of the tunneling oscillations evolves from the two-state to the three-state one. Our calculations demonstrate that the change in the underlying mechanism leads to the dramatic enhancement of the dynamical tunneling. Moreover, a complete suppression of the dynamical tunneling can also be achieved in the cold atom setup.