Progress in laser-driven proton acceleration requires increasing the proton maximum energy and laser-to-proton conversion efficiency, while reducing the divergence of the proton beam. However, achieving all these qualities simultaneously has proven challenging experimentally, the increase in beam energy often coming at the cost of beam quality. Numerical simulations suggest that coupling multi-PW laser pulses with ultrathin foils could offer a route for such simultaneous improvement. Yet experimental investigations have been limited by the scarcity of such lasers and the need for very stringent temporal contrast conditions to prevent premature target expansion before the pulse maximum. Here, combining the newly commissioned Apollon laser facility that delivers high-power ultrashort (∼24fs) pulses, with a double plasma mirror scheme to enhance its temporal contrast, we demonstrate the generation of up to 35 MeV protons with only 5 J of laser energy. This approach also achieves improved laser-to-proton energy conversion efficiency, reduced beam divergence, and optimized spatial beam profile. Therefore, despite the laser energy losses induced by the plasma mirror, the proton beams produced by this method are enhanced on all accounts compared to those obtained under standard conditions. Particle-in-cell simulations reveal that this improvement results from the intensified electrostatic field caused by a more compact hot-electron distribution.