  The electric conductivity of Dirac electrons in the organic conductor
$\alpha$-(BEDT-TTF)$_2$I$_3$ [BEDT-TTF = bis(ethylenedithio)tetrathiafulvalene]
under pressure has been examined using a two-dimensional tight-binding (TB)
model with both impurity and electron--phonon (e--p) scatterings. We study an
anomalous temperature dependence of the conductivity, which shows a crossover
from $\sigma_{x} < \sigma_{y}$ at low temperatures [region (I)] to $\sigma_{x}
> \sigma_{y}$ at high temperatures [region (II)]. $\sigma_y$ and $\sigma_x$ are
diagonal conductivities parallel and perpendicular to a stacking axis of
molecules, respectively. The effect of Dirac cone tilting is dominant in region
(I), whereas the anisotropy of the velocity of the Dirac cone is dominant in
region (II). Such behavior is further examined by calculating the deviation of
principal axes due to the off-diagonal conductivity $\sigma_{xy}$ and a nearly
constant conductivity at high temperatures due to the e--p scattering, which is
the extension of the previous result of the simple two-band model [Phys. Rev. B
{\bf 98},161205 (2018)]. The relevance to experiments of organic conductors is
discussed.
