  We revisit the $J/\psi$ pair production issue at the Fermilab Tevatron Run II
with the center-of-mass energy $\sqrt{s}=1.96 \mathrm{TeV}$. Both color-singlet
and -octet production mechanisms are considered. Our result shows that the
transverse momentum($p_T$) scaling behaviors of double $J/\psi$ differential
cross sections in color-singlet and -octet deviate distinctively from each
other while $p_T$ is larger than $8 \mathrm{GeV}$, and with the luminosity of
${5 \mathrm{fb^{-1}}}$ the $J/\psi$ pair events from color-singlet scheme are
substantially measurable in Tevatron experiments, even with certain lower
transverse momentum cut. Hence the Tevatron is still possibly a platform to
check the heavy quarkonium production mechanism.
