Laser-driven ion accelerators can deliver high-energy, high-peak current beams and are thus attracting attention as a compact alternative to conventional accelerators. However, achieving sufficiently high energy levels suitable for applications such as radiation therapy remains a challenge for laser-driven ion accelerators. Here we generate proton beams with a spectrally separated high-energy component of up to 150 MeV by irradiating solid-density plastic foil targets with ultrashort laser pulses from a repetitive petawatt laser. The preceding laser light heats the target, leading to the onset of relativistically induced transparency upon main pulse arrival. The laser peak then penetrates the initially opaque target and triggers proton acceleration through a cascade of different mechanisms, as revealed by three-dimensional particle-in-cell simulations. The transparency of the target can be used to identify the high-performance domain, making it a suitable feedback parameter for automated laser and target optimization to enhance stability of plasma accelerators in the future.