Ag nanocubes (NCs)/4-aminothiophenol (p-ATP)/smooth platinum (Pt) film (Ag-NCs @ p-ATP/Pt) sandwich structure is created for surface enhanced Raman scattering (SERS). The proposed sandwich structure is shown to exhibit better performance than the Ag-NCs only as SERS substrate. The dependence of the Raman signal intensity on the thickness of the Pt films is examined. It is shown that the Raman signal increases with the thickness of the Pt films from 42 to 90 nm, suggesting the electromagnetic coupling of the localized surface plasmons of the Ag-NCs with the surface plasmon polaritons of the underneath Pt film, which is confirmed by our numerical simulations. The SERS enhancement factor in Ag-NCs @ p-ATP/Pt is estimated to be (4.1 ± 0.2) × 106 for a Pt film of 90 nm.