Optical bound states in the continuum (BICs) are spatially localized states with vanishing radiation, despite their energy embedded in the continuum spectrum of the environment. They are expected to greatly enhance light–matter interaction due to their long lifetime and high quality factor. However, the BICs in all-dielectric structures generally exhibit large mode volumes and their properties are difficult to manipulate. In this paper, we propose a metal–dielectric hybrid nanostructure where a silver film is inserted into the silicon (Si) substrate under the Si nanopillar array. We show that symmetry-protected BIC in this system can couple with surface plasmon polaritons (SPPs) to form a hybridized mode. Compared with previous symmetry-protected BICs in all-dielectric structures, the SPP-coupled BIC has a significantly decreased mode volume, and its corresponding electric field is strongly localized below the Si nanopillars. We also show that the SPP mode makes the original polarization-independent symmetry-protected BIC become polarization-dependent. In addition, we demonstrate that the silver film in the considered structure can induce a metal mirror effect. The destructive interference between the magnetic dipole inside the Si nanopillars and the mirror magnetic dipole in the silver film can lead to the formation of accidental BICs. Our hybrid structure provides a versatile platform for the manipulation of light–matter interaction in the nanoscale.