  In this paper, we implement a generalised pseudo-Newtonian potential to study
the off-equatorial orbits inclined at a certain angle with the equatorial plane
around Schwarzschild and Kerr-like compact object primaries surrounded by a
dipolar halo of matter. The chaotic dynamics of the orbits are detailed for
both non-relativistic and special-relativistic test particles. The dependence
of the degree of chaos on the Kerr parameter $a$ and the inclination angle $i$
is established individually using widely used indicators, such as the
Poincar\'e Maps and the Maximum Lyapunov Exponents. Although the orbits'
chaoticity has a positive correlation with $i$, the growth in the chaotic
behaviour is not systematic. There is a threshold value of the inclination
angle $i_{\text{c}}$, after which the degree of chaos sharply increases. On the
other hand, the chaoticity of the inclined orbits anti-correlates with $a$
throughout its entire range. However, the negative correlation is systematic at
lower values of the inclination angle. At higher values of $i$, the degree of
chaos increases rapidly below a threshold value of the Kerr parameter,
$a_{\text{c}}$. Above this threshold value, the correlation becomes weak.
Furthermore, we establish a qualitative correlation between the threshold
values and the overall chaoticity of the system. The studies performed with
different orbital parameters and several initial conditions reveal the
intricate nature of the system.
