  After the investigation of the congruent and non-congruent hyperball packings
related to doubly truncated Coxeter orthoscheme tilings \cite{SzJ1}, we
consider the corresponding covering problems. In \cite{MSSz} the authors gave a
partial classification of supergroups of some hyperbolic space groups whose
fundamental domains will be integer parts of truncated tetrahedra, and
determined the optimal congruent hyperball packing and covering configurations
belonging to some of these classes.
  In this paper we compliment these results with the investigation of the
non-congruent covering cases, and the remainig congruent cases. We prove, that
between congruent and non-congruent hyperball coverings the thinnest belongs to
the $\{7,3,7\}$ Coxeter tiling with density $\approx 1.26829$. This covering
density is smaller than the conjectured lower bound density of L.~Fejes~T\'oth
for coverings with balls and horoballs.
  We also study the local packing arrangements related to $\{u,3,7\}$ $(6< u <
7, ~ u\in \mathbb{R})$ doubly truncated orthoschemes and the corresponding
hyperball coverings. We prove, that these coverings are achieved their minimum
density at parameter $u\approx 6.45953$ with covering density $\approx 1.26454$
which is smaller then the above record-small density, but this hyperball
arrangement related to this locally optimal covering can not be extended to the
entire $\mathbb{H}^3$. Moreover we see, that in the hyperbolic plane
$\mathbb{H}^2$ the universal lower bound of the congruent circle, horocycle,
hypercycle covering density $\frac{\sqrt{12}}{\pi}$ can be approximated
arbitrarily well also with non-congruent hypercycle coverings generated by
doubly truncated Coxeter orthoschemes.
