  We study the magnetization process of the $S=1$ Heisenberg model on a two-leg
ladder with further neighbor spin-exchange interaction. We consider the
interaction that couples up to next-nearest neighbor rungs. We find an exactly
solvable point at which the magnetization plateaus take the multiples of $1/6$
and draw a general phase diagram for weaker next-nearest neighbor interaction
using the density matrix renormalization group method. We observe a sizable
parameter region where magnetization plateaus at multiples of $1/6$ and $1/4$
coexist.
