It was proved [Opt. Lett. 30, 720 (2005)] that the deposition of an overlay of higher refractive index than that of the cladding on a long-period fiber grating (LPFG) causes large shifts in the attenuation bands induced by the grating. The result is an enhancement of the sensitivity of the LPFG to variations in the ambient and overlay refractive indices or the overlay thickness. The limitation of the previous design to materials with higher refractive indices than that of the cladding of the LPFG is overcome with a five-layer model. To this purpose, a first overlay of higher refractive index than that of the cladding of the LPFG will enhance the sensitivity of the device to variations in the refractive index of a second overlay of lower refractive index than that of the cladding of the LPFG. Moreover, it is proved that, if the second overlay is thick enough, its behavior resembles that of an infinite layer.