Transformer-based automatic speech recognition (ASR) systems have shown their success in the presence of large datasets. But, in medical research, we have to create ASR for the non-typical population, i.e. pre-school children with speech disorders, with small training dataset. To increase training efficiency on small datasets, we optimize the architecture of Wav2Vec 2.0, a variation of Transformer, through analyzing its pre-trained model's block-level attention pattern. We show that block-level patterns can serve as an indicator for narrowing down the optimization direction. To ensure the reproducibility of our experiments, we leverage Librispeech-100-clean as training data to simulate the limited data condition. We leverage two techniques, local attention mechanism and cross-block parameter sharing, with counter-intuitive configurations. Our optimized architecture outperforms the vanilla architecture about 1.8% absolute word error rate (WER) on dev-clean and 1.4% on test-clean.