Laser-driven ion accelerators can deliver high-energy, high peak current beams from relativistic laser plasmas formed in solid-density materials [1, 2]. This innovative concept attracts a lot of attention for various multidisciplinary applications as a compact alternative to conventional accelerators [3]. However, achieving energy levels suitable for applications such as radiation therapy remains a challenge for laser-driven ion accelerators. Here, we report on experimental generation of plasma-accelerated proton beams with a spectrally separated high-energy component of up to 150MeV by irradiating solid-density plastic foil targets with ultrashort laser pulses from a repetitive Petawatt laser. Three-dimensional particle-in-cell simulations reveal that the observed beam parameters result from cascaded acceleration regimes that occur at the onset of relativistically induced transparency. The ultrashort pulse duration allows a rapid sequence of these regimes at highest intensity, enabling proton acceleration to unprecedented energy levels. Target transparency was identified to discriminate the high-performance domain of the acquired data set, making it a suitable feedback parameter for automated laser and target optimisation to enhance stability of plasma accelerators in the future. Ultimately, our results encourage further exploration and application of laser-driven plasmas as compact proton accelerators in the multi-100MeV range.