This article proposes a novel technique for accelerating the convergence of the previously published norm-optimal iterative learning control (NOILC) methodology. The basis of the results is a formal proof of an observation made by D.H. Owens, namely that the NOILC algorithm is equivalent to a successive projection algorithm between linear varieties in a suitable product Hilbert space. This leads to two proposed accelerated algorithms together with well-defined convergence properties. The results show that the proposed accelerated algorithms are capable of ensuring monotonic error norm reductions and can outperform NOILC by more rapid reductions in error norm from iteration to iteration. In particular, examples indicate that the approach can improve the performance of NOILC for the problematic case of non-minimum phase systems. Realisation of the algorithms is discussed and numerical simulations are provided for comparative purposes and to demonstrate the numerical performance and effectiveness of the proposed methods.