We present the design of a new architecture of three-terminal (3-T) photovoltaic tandem solar cells. It combines an interdigitated back contacted (IBC) bottom lateral subcell with a heterojunction vertical top cell. In this concept, the two subcells can work independently and there is no need for tunnel junctions. It is particularly well suited to silicon back contact subcells and to various types of top cell materials from III-V compounds or perovskites. The working principle is detailed here using as an example a p/i III-V front stack onto n-type silicon IBC bottom cell. We perform two-dimensional (2D) modeling using realistic material input parameters and show how interface bandgap engineering can improve the tandem cell efficiency up to a reachable 35% value. The proposed cell concept named bandgap engineered smart silicon three-terminal (BESTT) cell can be realized with less technological steps and at a lower cost compared with the conventional four-terminal process.