  We perform a complete study of prompt double $J/\psi$ hadroproduction at
leading order in the nonrelativistic-QCD factorization framework by including
all possible pairings of the $c\bar{c}$ Fock states ${}^1\!S_0^{[8]}$,
${}^3\!S_1^{[1,8]}$, and ${}^3\!P_J^{[1,8]}$ with $J=0,1,2$. We find that the
${}^1\!S_0^{[8]}$ and ${}^3\!P_J^{[8]}$ channels of $J/\psi$ and $\psi^\prime$
production and the ${}^3\!P_J^{[1]}$ and ${}^3\!S_1^{[8]}$ channels of
$\chi_{cJ}$ production, which have been overlooked so far, greatly dominate at
large invariant masses and rapidity separations of the $J/\psi$ pair, and that
their inclusion nearly fills the large gap between previous incomplete
predictions within the color-singlet model and the recent measurement by the
CMS Collaboration at the CERN LHC, leaving room for next-to-leading-order
corrections of typical size.
