|Penumbral jets are bright transient narrow upflows in the chromosphere of sunspot penumbrae. They have been proposed to come from component (acute angle) reconnection of the field in spines with that in interspines and to possibly contribute to transition-region and coronal heating above sunspots. Based on the recently revealed internal structure of sunspot penumbral filaments, both in observations and magnetohydrodynamic simulations, in which filaments are found to be magnetized stretched granule-like convection cells, we propose that most penumbral jets form as a result of reconnection of the opposite polarity field at the sides of filaments with spine field. By using Hinode (SOT/FG), Hi-C and SDO/AIA observations of a sunspot penumbra, we find that these jets do not significantly directly heat the corona above sunspots, except a few of them showing up in the AIA 1600 images. We discovered a few locations of mixed polarity flux (in Stokes-V images) at tails of penumbral filaments in the sunspot penumbra where larger jets were produced repetitively. These larger jets show clear signatures in the transition region but their coronal contribution remains ambiguous. Our proposed modified picture of formation of penumbral jets includes the formation of the larger penumbral jets that form at filament tails.