It was proved that the deposition of an overlay material onto a long-period fiber grating causes important shifts in the wavelengths of the typical attenuation bands that are caused by coupling between cladding and core modes [Opt. Lett. 27, 682 (2002)]. A theoretical model for analyzing a multilayer cylindrical waveguide is presented that permits the phenomenon to be understood and predicted. An overlay of higher refractive index than the cladding starts to guide a mode if a certain thickness value is exceeded. This causes large shifts in the resonance wavelength induced by the grating. One important application of this phenomenon to sensors is enhancement of the sensitivity of a long-period fiber grating to ambient conditions. Theoretical results are corroborated with experimental ones obtained by electrostatic self-assembly.