Cylindrically symmetric hybrid fiber proposed by our group recently allow one to achieve high anomalous dispersion (13-100 ps/nm/km) in the 1 µm spectral range. Refractive index of such fibers consists of one or two high index ring layer surrounding the core and depressed layer placed outside the ring layer. The drawback of the realized designs was presence of modes, localized in high-index rings, which tails nevertheless spread over the hybrid fiber core. Excitation of the operating hybrid mode LP 03 in the two high-index ring design (or LP 02 in the one high-index layer design) was accompanied by weak excitation (10-20% of total propagating power) of the modes, located in the high-index rings, that has non-zero electric field at fiber axis (mainly LP 02 mode for two high-index-layers design and LP 01 for one high-index layer design).