We introduce and study a new family of pseudo‐Riemannian metrics g̃λμν on the anti‐de Sitter three‐space H13. These metrics will be called “of Kaluza‐Klein type” , as they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle T1double-struckH2(κ). For any choice of three real parameters λ,μ,ν≠0, the pseudo‐Riemannian manifold ()double-struckH13,trueg̃λμν is homogeneous. Moreover, we shall introduce and study some natural almost contact and paracontact structures (φ,ξ,η), compatible with g̃λμν, such that (φ,ξ,η,trueg̃λμν) is a homogeneous almost contact (respectively, paracontact) metric structure. These structures will be then used to show the existence of a three‐parameter family of homogeneous metric mixed 3‐structures on the anti‐de Sitter three‐space.