A laser-Compton backscattering beam, which we call a `Laser-Electron Photon' beam, was upgraded at the LEPS beamline of SPring-8. We accomplished the gains in backscattered photon beam intensities by factors of 1.5--1.8 with the injection of two adjacent laser beams or a higher power laser beam into the storage ring. The maximum energy of the photon beam was also extended from 2.4 GeV to 2.9 GeV with deep-ultraviolet lasers. The upgraded beams have been utilized for hadron photoproduction experiments at the LEPS beamline. Based on the developed methods, we plan the simultaneous injection of four high power laser beams at the LEPS2 beamline, which has been newly constructed at SPring-8. As a simulation result, we expect an order of magnitude higher intensities close to 10$^7$ sec$^{-1}$ and 10$^6$ sec$^{-1}$ for tagged photons up to 2.4 GeV and 2.9 GeV, respectively. ; Comment: 31 pages, 9 figures, Submitted to Nuclear Instruments and Methods in Physics Research Section A on 29 May 2013