Two- and three-dimensional topological insulators are the key materials for the future nanoelectronic and spintronic devices and quantum computers. By means of angle- and spin-resolved photoemission spectroscopy we study the electronic and spin structure of the Bi-bilayer/3D topological insulator in quantum tunneling regime formed under the short annealing of Bi2Te2.4Se0.6. Owing to the temperature-induced restructuring of the topological insulator's surface quintuple layers, the hole-like spin-split Bi-bilayer bands and the parabolic electronic-like state are observed instead of the Dirac cone. Scanning Tunneling Microscopy and X-ray Photoemission Spectroscopy measurements reveal the appearance of the Bi2 terraces at the surface under the annealing. The experimental results are supported by density functional theory calculations, predicting the spin-polarized Bi-bilayer bands interacting with the quintuple-layers-derived states. Such an easily formed heterostructure promises exciting applications in spin transport devices and low-energy electronics.