For a nanowire quantum dot with the confining potential modeled by both the infinite and the finite square wells, we obtain exactly the energy spectrum and the wave functions in the strong spin-orbit coupling regime. We find that regardless of how small the well height is, there are at least two bound states in the finite square well: one has the $sigma^{x}mathcal{P}=-1$ symmetry and the other has the $sigma^{x}mathcal{P}=1$ symmetry. When the well height is slowly tuned from large to small, the position of the maximal probability density of the first excited state moves from the center to $x e0$, while the position of the maximal probability density of the ground state is always at the center. A strong enhancement of the spin-orbit effect is demonstrated by tuning the well height. In particular, there exists a critical height $V^{c}_{0}$, at which the spin-orbit effect is enhanced to maximal.
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