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We present a hybrid GPU-CPU implementation of a reordering strategy for per-muting elements to make the half-bandwidth of a sparse matrix lower. This algorithm, based on work by Cuthill and McKee [4], is a critical component of the SPIKE-based preconditioner provided by the Spike::GPU library [3]. We discuss the three stages of the unordered implementation of the Cuthill-McKee (CM) algorithm and compare our implementation against both a self-implemented CPU CM algorithm and the MC60 algo-rithm provided by the HSL library [1], which also targets reducing the half-bandwidth. Numerical experiments, performed on around 130 matrices arising in various engineer-ing and scientific applications, indicate that our implementation almost obtains similar or narrower reordered matrix, and for large matrices with dimension over 106 and num-ber of nonzero entries over 107, our implementation is more efficient than both other implementations