We are presenting a theoretical study of light transmission through a slab of hyperbolic metamaterial composed of gold rods, with chiral inclusions, thus possessing both hyperbolic and chiral properties. 4 × 4 transfer matrix formalism was used for solution of a wave transmission through a slab of a chiral hyperbolic material. We have shown that circular dichroism (CD) can be enhanced by several orders of magnitude in epsilon-near-zero (ENZ) mode when the diagonal component of the permittivity tensor corresponding to the normal to interface coordinate tends to zero. The necessary condition of the enhancement is a non-zero light incidence angle. This effect results from a considerable wavelength shortening in normal direction that increases the interaction between light and matter. We have shown that the increase of the incidence angle leads to the increase of dichroism in vicinity of ENZ wavelength range and the dichroism can change sign in this resonance area. The dichroism becomes huge at large incidence angles (θ > 30°) and for each incidence angle the maximal enhancement is observed within a certain wavelength range (in vicinity of-near-zero). In opposite to discussion of the enhancement of the pseudochirality reported previously in the literature, our study relates to the true chirality. We expect these results to be of interest to researchers engaged in chemical composition of organic substances by retrieval of effective parameters from measurements of the CD. © 2019 IOP Publishing Ltd.