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Designing a speech quality assessment (SQA) system for estimating mean-opinion-score (MOS) of multi-rate speech with varying sampling frequency (16-48 kHz) is a challenging task. The challenge arises due to the limited availability of a MOS-labeled training dataset comprising multi-rate speech samples. While self-supervised learning (SSL) models have been widely adopted in SQA to boost performance, a key limitation is that they are pretrained on 16 kHz speech and therefore discard high-frequency information present in higher sampling rates. To address this issue, we propose a spectrogram-augmented SSL method that incorporates high-frequency features (up to 48 kHz sampling rate) through a parallel-branch architecture. We further introduce a two-step training scheme: the model is first pre-trained on a large 48 kHz dataset and then fine-tuned on a smaller multi-rate dataset. Experimental results show that leveraging high-frequency information overlooked by SSL features is crucial for accurate multi-rate SQA, and that the proposed two-step training substantially improves generalization when multi-rate data is limited.