  The spin-1 Haldane chain is an example of the symmetry-protected-topological
(SPT) phase in one dimension. Experimental realization of the spin chain
materials usually involves both the uniaxial-type, $D(S^z)^2$, and the
rhombic-type, $E[(S^x)^2-(S^y)^2]$, single-ion anisotropies. Here, we provide a
precise ground-state phase diagram for spin-1 Haldane chain with these
single-ion anisotropies. Using quantum numbers, we find that the $\mathbb{Z}_2$
symmetry breaking phase can be characterized by double degeneracy in the
entanglement spectrum. Topological quantum phase transitions take place on
particular paths in the phase diagram, from the Haldane phase to the
Large-$E_x$, Large-$E_y$, or Large-$D$ phases. The topological critical points
are determined by the level spectroscopy method with a newly developed parity
technique in the density matrix renormalization group [Phys. Rev. B 86, 024403
(2012)], and the Haldane-Large-$D$ critical point is obtained with an
unprecedentedly precision, $(D/J)_c=0.9684713(1)$. Close to this critical
point, a small rhombic single-ion anisotropy $|E|/J\ll1$ can destroy the
Haldane phase and bring the system into a $y$-N\'eel phase. We propose that the
compound [Ni(HF$_2$)(3-Clpy)$_4$]BF$_4$ is a candidate system to search for the
$y$-N\'eel phase.
