Convolutional neural networks (CNNs) have become an increasingly popular tool for brain lesion segmentation in recent years due to its accuracy and efficiency. However, CNN-based brain lesion segmentation generally requires a large amount of annotated training data, which can be costly for medical imaging. In many scenarios, only a few annotations of brain lesions are available. One common strategy to address the issue of limited annotated data is to transfer knowledge from a different yet relevant source task, where training data is abundant, to the target task of interest. Typically, a model can be pretrained for the source task, and then fine-tuned with the scarce training data associated with the target task. However, classic fine-tuning tends to make small modifications to the pretrained model, which could hinder its adaptation to the target task. Fine-tuning with increased model capacity has been shown to alleviate this negative impact in image classification problems. In this work, we extend the strategy of fine-tuning with increased model capacity to the problem of brain lesion segmentation, and then develop an advanced version that is better suitable for segmentation problems. First, we propose a vanilla strategy of increasing the capacity, where, like in the classification problem, the width of the network is augmented during fine-tuning. Second, because unlike image classification, in segmentation problems each voxel is associated with a labeling result, we further develop a spatially adaptive augmentation strategy during fine-tuning. Specifically, in addition to the vanilla width augmentation, we incorporate a module that computes a spatial map of the contribution of the information given by width augmentation in the final segmentation. For demonstration, the proposed method was applied to ischemic stroke lesion segmentation, where a model pretrained for brain tumor segmentation was fine-tuned, and the experimental results indicate the benefit of our method.