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The most time-consuming part of radiative transfer cosmological simulations for the Epoch of Reionisation is the ray-tracing algorithm. This involves computing the hydrogen column density along the path of the ionising photons. In this work, we present an updated version of pyC2Ray1, a massively parallel ray-tracing and chemistry library that supersedes C2Ray which has been extensively employed in reionisation simulations and often requires millions of CPU-core hours across several thousand computing nodes on high-performance computers to simulate large-scale volumes. pyC2Ray implements the Accelerated Short-characteristics Octahedral RAy-tracing (ASORA) algorithm, a novel ray-tracing method explicitly designed for multi-source ray-tracing on GPUs. The algorithm is written in C++/CUDA and wrapped in a Python interface that enables easy, customised code usage without compromising computational efficiency. We demonstrate that the ASORA updates introduced in this work lead to a speed-up factor of 800 compared to C2Ray on a multi-core CPU. It is an indication that ray-tracing algorithms for cosmological simulations strongly benefit from GPU-acceleration and specialised software design development to achieve computational efficiency and physical accuracy.