This note aims to present further results on the accelerated exponential Euler method proposed in Jentzen & Kloeden (2009). In contrast to very restrictive assumptions made there, we reformulate appropriate conditions on the drift coefficient of SPDEs to include a large class of nonlinear Nemytskii operators. In our setting, the method achieves the convergence order in time of 1 2− ϵ for arbitrarily small ϵ> 0 in the case of space–time white noise. For the trace-class noise case, multiple spatial dimensions are allowed and an optimal convergence rate is attained based on optimal regularity results of the mild solution, which improves the corresponding convergence results in Jentzen et al.(2011).