We reexamine the influence of the flexoelectricity on the electroconvection (EC), the effect which has previously been studied in the conductive EC regime only. Now we extend our studies to the dielectric EC, and to the parameter range for which the standard model of EC excludes the existence of the instability. First, we demonstrate that inclusion of flexoelectricity into the standard model of EC provides a finite EC threshold even for this "forbidden'' material parameter range (in accordance with the experiments). Second, we show that flexoelectricity considerably decreases the threshold voltage for the dielectric regime. Finally, we present a novel frequency dependence of the threshold voltage for all EC patterns in the case when the period of the driving frequency becomes comparable with the director relaxation time.