Oscillating-foil turbines have already proven to achieve efficiencies higher than 40% when operating at heaving amplitudes of the order of one chord with sinusoidal motions. In this 2D numerical parametric study, the energy extraction of an oscillating foil at a Reynolds number of 500,000 is maximized by prescribing large heaving amplitudes in combination with a modified pitching motion that imposes a sinusoidal evolution of the angle of attack. This allows to maintain a high efficiency of 44% for heaving amplitudes as large as 15 chords, meaning that a single large-amplitude turbine could efficiently exploit the same extraction window as about 10 smaller turbines having a standard one-chord heaving amplitude. By further modifying the pitching function to maintain a high angle of attack over a larger portion of the cycle, it is also possible to reach efficiencies as high as 49%. It is important to note that these impressive efficiencies are achieved at the cost of a significant energy input for the reversal of the foil at every half cycle, representing a potential practical challenge for a pitching controller.