A two-stage lightweight online dereverberation algorithm for hearing devices is presented in this paper. The approach combines a multi-channel multi-frame linear filter with a single-channel single-frame post-filter. Both components rely on power spectral density (PSD) estimates provided by deep neural networks (DNNs). By deriving new metrics analyzing the dereverberation performance in various time ranges, we confirm that directly optimizing for a criterion at the output of the multi-channel linear filtering stage results in a more efficient dereverberation as compared to placing the criterion at the output of the DNN to optimize the PSD estimation. More concretely, we show that training this stage end-to-end helps further remove the reverberation in the range accessible to the filter, thus increasing the \textit{early-to-moderate} reverberation ratio. We argue and demonstrate that it can then be well combined with a post-filtering stage to efficiently suppress the residual late reverberation, thereby increasing the \textit{early-to-final} reverberation ratio. This proposed two stage procedure is shown to be both very effective in terms of dereverberation performance and computational demands, as compared to e.g. recent state-of-the-art DNN approaches. Furthermore, the proposed two-stage system can be adapted to the needs of different types of hearing-device users by controlling the amount of reduction of early reflections.