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After the investigation of the congruent and non-congruent hyperball packingsrelated to doubly truncated Coxeter orthoscheme tilings \cite{SzJ1}, weconsider the corresponding covering problems. In \cite{MSSz} the authors gave apartial classification of supergroups of some hyperbolic space groups whosefundamental domains will be integer parts of truncated tetrahedra, anddetermined the optimal congruent hyperball packing and covering configurationsbelonging to some of these classes.  In this paper we compliment these results with the investigation of thenon-congruent covering cases, and the remainig congruent cases. We prove, thatbetween congruent and non-congruent hyperball coverings the thinnest belongs tothe $\{7,3,7\}$ Coxeter tiling with density $\approx 1.26829$. This coveringdensity is smaller than the conjectured lower bound density of L.~Fejes~T\'othfor 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 correspondinghyperball coverings. We prove, that these coverings are achieved their minimumdensity at parameter $u\approx 6.45953$ with covering density $\approx 1.26454$which is smaller then the above record-small density, but this hyperballarrangement related to this locally optimal covering can not be extended to theentire $\mathbb{H}^3$. 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 approximatedarbitrarily well also with non-congruent hypercycle coverings generated bydoubly truncated Coxeter orthoschemes.

Coverings with congruent and non-congruent hyperballs generated by doubly truncated Coxeter orthoschemes