  As a continuation of our previous work ({\it Phys. Rev. A68, 012504 (2003)})
an accurate study of the lowest $1\si_g$ and the low-lying excited $1\si_u$,
$2\si_g$, $1\pi_{u,g}$, $1\de_{g,u}$ electronic states of the molecular ion
$H_2^+$ is made. Since the parallel configuration where the molecular axis
coincides with the magnetic field direction is optimal, this is the only
configuration which is considered. The variational method is applied and the
{\it same} trial function is used for different magnetic fields. The magnetic
field ranges from $10^9 G$ to $4.414 \times 10^{13} G$ where non-relativistic
considerations are justified. Particular attention is paid to the $1\si_u$
state which was studied for an arbitrary inclination. For this state a
one-parameter vector potential is used which is then variationally optimized.
