  The additional resonant contribution to the potential model is examined in
$\alpha$+$^{12}$C elastic scattering and the low-energy
$^{12}$C($\alpha$,$\gamma$)$^{16}$O reaction. The excitation function of
elastic scattering below $E_{c.m.}= 5$ MeV seems to be reproduced by the
potential model satisfactorily, and it is not profoundly disturbed by the
additional resonances. The weak coupling is good enough to describe the
$^{16}$O structure in the vicinity of the $\alpha$-particle threshold,
especially below $E_{c.m.}= 8$ MeV, corresponding to the excitation energy $E_x
\approx 15$ MeV. The additional resonances give the complement of the
astrophysical $S$-factors from the simple potential model. The $S$-factor of
$^{12}$C($\alpha$,$\gamma$)$^{16}$O at $E_{c.m.}=300$ keV is dominated by the
$E$2 transition, which is enhanced by the subthreshold 2$^+_1$ state at $E_x=
6.92$ MeV. The contribution from the subthreshold 1$^-_1$ state at $E_x= 7.12$
MeV is predicted to be small. The additional resonances do not give the large
contribution to the thermonuclear reaction rates of
$^{12}$C($\alpha$,$\gamma$)$^{16}$O at helium burning temperatures.
