  Adopting the complete $\alpha_s^4$ approach of the perturbative QCD (pQCD)
and updated parton distribution functions, we have estimated the hadronic
production of $P$-wave excited $B_c$-states ($B_{cJ,L=1}^*$). In the estimate,
special care on the relation of the production amplitude to the derivative of
wave function at origin of the potential model is payed. For experimental
references, main uncertainties are discussed, and the total cross sections and
the distributions of the production with reasonable cuts at the energies of
Tevatron and LHC are computed and presented. The results show that $P$-wave
production may contribute to the $B_c$-meson production indirectly by a factor
about 0.5 of the direct production, and with such a big cross section, it is
worth further to study the possibility to observe the $P$-wave production
itself experimentally.
