In the pursuit of designing highly effective decoders for short LDPC codes nearing maximum likelihood performance, we employ a relayed decoding strategy. A neural min-sum decoder initiates the decoding process, and its errors undergo postprocessing through an adaptive ordered statistics decoding. Several key initiatives supporting the latter are emphasized. Firstly, soft information at each iteration of the neural min-sum decoder is gathered and input into a convolutional neural network to enhance bit reliability estimates. This process identifies error-prone bits, either excluding them from the most reliable basis or concentrating them at the forefront, both advantageous for the adaptive ordered statistical decoding. Additionally, a decoding path, comprising a list of order patterns, directs the postprocessing process. Adjustable path length and refined constraints on associated order patterns offer diverse means of complexity management. Simultaneously, a novel auxiliary criterion is introduced to significantly reduce the list size of codeword candidates in the adaptive ordered statistics decoding. Extensive experimental results and complexity analysis validate the serial architecture, equipped with these innovations, as a formidable contender against state-of-the-art decoders for short LDPC codes.

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