Novel imaging methods show that the mobile dopants in optimum doped La 2 CuO 4+y (LCO) get self-organized, instead of randomly distributed, to form an inhomogeneous network of nanoscale metallic puddles with ordered oxygen interstitials interspersed with oxygen-depleted regions. These puddles are expected to be metallic, being far from half filling because of high dopant density, and to sustain superconductivity having a size in the range 5-20 nm. However, the electronic structure of these heavily doped metallic puddles is not known. In fact the rigid-band model fails because of ordering of dopants and supercell calculations are required to obtain the Fermi surface reconstruction. We have performed advanced band calculations for a large supercell La 16 Cu 8 O 32+N where N = 1 or 2 oxygen interstitials form rows in the spacer La 16 O 16+N layers intercalated between the CuO 2 layers as determined by scanning nano x-ray diffraction. The additional occupied states made by interstitial oxygen orbitals sit well below the Fermi level (E F ) and lead to hole doping as expected. The unexpected results show that in the heavily doped puddles the altered Cu(3d)-O(2p) band hybridization at E F induces a multiband electronic structure with the formation of multiple Fermi surface spots: (a) Small gaps appear in the folded Fermi surface, (b) three minibands cross E F with reduced Fermi energies of 60, 150, and 240 meV, respectively, (c) the density of states and band mass at E F show substantial increases, and (d) spin-polarized calculations show a moderate increase of antiferromagnetic spin fluctuations. All calculated features are favorable to enhance superconductivity; however, the comparison with experimental methods probing the average electronic structure of cuprates will require the description of the electronics of a network of multigap superconducting puddles.