  We extend our previous work on tests of the Kerr black hole hypothesis using
X-ray reflection spectroscopy and we infer observational constraints on the
Johannsen deformation parameter $\epsilon_3$. We analyze a NuSTAR observation
of the stellar-mass black hole in GS 1354-645, Suzaku data of the supermassive
black holes in Ark 564, 1H0419-577, and PKS 0558-504, and some simultaneous
XMM-Newton and NuSTAR observations of the supermassive black hole in
MCG-6-30-15. All our measurements of $\epsilon_3$ are consistent with the Kerr
metric and we find quite strong constraints from Ark 564 and MCG-6-30-15. We
discuss the implications of our results and the next steps to improve our
tests.
