Cloaking devices enabling to hide objects from external observers have numerous potential applications but still face challenges in realization, especially in the visible spectral range. In particular, inherent losses and extreme parameters of metamaterials required for the cloaking are the limiting factors. Here, we propose and demonstrate numerically nearly per- fect concealing of objects in ENZ-metamaterials (homogeneous and layered realizations) acting alignment-free scattering suppression covers. The suppression of visibility relies on the recon- struction of a plane wavefront after scattering of light by an object placed inside an anisotropic metamaterial with vanishing permittivity along one of the major crystallographic directions (uniaxial ENZ metamaterial). We also consider a realistic realization of the metamaterial as a metal-dielectric multilayer and demonstrate the cloaking in forward and backward directions with perfectly preserved wavefronts and less than 10% absolute intensity change.