In this article, we deal with new properties of a Solid Core Photonic Bandgap (SC-PBGF) fiber with intersticial air holes (IAHs) in its transverse structure. It has been shown recently, that IAH enlarges its bandgaps (BG), compared to what is observed in a regular SC-PBGF. We shall describe the mechanisms that account for this BG opening, which has not been explained in detail yet. It is then interesting to discuss the role of air holes in the modification of the Bloch modes, at the boundaries of the BG. In particular, we will use a simple method to compute the exact BG diagrams in a faster way, than what is done usually, drawing some parallels between structured fibers and physics of photonic crystals. The very peculiar influence of IAHs on the upper/lower boundaries of the bandgaps will be explained thanks to the difference between mode profiles excited on both boundaries, and linked to the symmetry / asymmetry of the modes. We will observe a modification of the highest index band (n(FSM)) due to IAHs, that will enable us to propose a fiber design to guide by Total Internal Reflection (TIR) effect, as well as by a more common BG confinement. The transmission zone is deeply enlarged, compared to regular photonic bandgap fibers, and consists in the juxtaposition of (almost non overlapping) BG guiding zones and TIR zone.