Let $(M,g,J)$ be an almost Hermitian manifold, then the tangent bundle $TM$ carries a class of naturally defined almost hyper-Hermitian structures $(G,J_1,J_2,J_3)$. In this paper we give conditions under which these almost hyper-Hermitian structures $(G,J_1,J_2,J_3)$ are locally conformal hyper-Kähler. As an application, a family of new hyper-structures is obtained on the tangent bundle of a complex space form. Furthermore, by restricting these almost hyper-Hermitian structures on the unit tangent sphere bundle $T_{1}M$, we obtain a class of almost...