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We theoretically detail and explain how to control unwanted excess gain for higher order modes in cladding-pumped large-core ytterbium-doped fiber amplifiers operating on the fundamental mode by using a small inner cladding and pumping close to the emission wavelength in a so-called tandem-pump configuration. Such excess gain can otherwise limit the core size. We study this experimentally by comparing the level of amplified spontaneous emission at the edge of the core in a conventional diode-pumped and tandem-pumped large-core ytterbium-doped multimode fiber. We also discuss the potential benefits of tandem pumping in reducing fiber photodarkening.