  We performed third-order electrical conductivity measurements on the organic
conductor $\alpha$-(BEDT-TTF)$_2$I$_3$ using an ac bridge technique sensitive
to nonlinear signals. Third-order conductance $G_3$ is clearly observed even at
low electric fields, and interestingly, $G_3$ is critically enhanced above the
charge-order transition temperature $T_{\rm CO}=136$~K. The observed frequency
dependence of $G_3$ is incompatible with a percolation model, in which a Joule
heating in a random resistor network is relevant to the nonlinear conduction.
We instead argue the nonlinearity of the relaxation time according to a
phenomenological model on the mobility in materials with large dielectric
constants, and find that the third-order conductance $G_3$ corresponds to the
third-order electric susceptibility $\chi_3$. Since the nonlinear
susceptibility is known as a probe for higher-order multipole ordering, the
present observation of the divergent behavior of $G_3$ above $T_{\rm CO}$
reveals an underlying quadrupole instability at the charge-order transition of
the organic system.
