We report a new transition scenario from a uniform state to the spatio-temporal chaos (STC) for the electroconvection in the homeotropically aligned nematics, in which a magnetic field is applied perpendicularly to the electric field. Above the Lifshitz frequency f(L) the amplitude of the electric field increasing, a stationary periodic rolls state becomes unstable, then evolves to STC-I (i.e., the defect chaos) at a certain electric field. At a higher electric field, the second stationary periodic rolls (i.e., abnormal rolls) are also found above a characteristic intensity of the magnetic field H*. At further higher electric field the second stationary periodic rolls state becomes unstable again, then evolves to more complicated STC (STC-II). Above H* the abnormal rolls instability corresponding to an azimuthal rotation of the director plays key roles on the pattern transitions between stationary and non-stationary states as well as between the two stationary ones. Based on the systematic study, a clear phase diagram was determined in the H-epsilon plane. These results are remarkably different from ones below f(L) previously reported.