  We make a detailed study on the typical production channel of double
charmoniums, $e^+e^-\to J/\psi+\eta_c$, at the center-of-mass collision energy
$\sqrt{s}=10.58$ GeV. The key component of the process is the form factor
$F_{\rm VP}(q^2)$, which has been calculated within the QCD light-cone sum
rules (LCSR). To improve the accuracy of the derived LCSR, we keep the $J/\psi$
light-cone distribution amplitude up to twist-4 accuracy. Total cross sections
for $e^+e^-\to J/\psi+\eta_c$ at three typical factorization scales are
$\sigma|_{\mu_s} = 22.53^{+3.46}_{-3.49}~{\rm fb}$, $\sigma|_{\mu_k} =
21.98^{+3.35}_{-3.38}~{\rm fb}$ and $\sigma|_{\mu_0} =
21.74^{+3.29}_{-3.33}~{\rm fb}$, respectively. The factorization scale
dependence is small, and those predictions are consistent with the BABAR and
Belle measurements within errors.
