We proposed and investigated a novel type of all-glass hybrid fiber where light is confined in the low-index core due to both total internal reflection and coherent Fresnel reflection (a photonic bandgap mechanism). The hybrid mode has an anomalous dispersion of 13 ps/(nm km) at 1064 nm and low loss (~6 dB/km), and it can be easily excited by splicing with a single-mode step-index fiber. The compression of positively chirped 8 ps pulses down to 330 fs was demonstrated with the fabricated hybrid fiber.