Traditional federated learning mainly focuses on parallel settings (PFL), which can suffer significant communication and computation costs. In contrast, one-shot and sequential federated learning (SFL) have emerged as innovative paradigms to alleviate these costs. However, the issue of non-IID (Independent and Identically Distributed) data persists as a significant challenge in one-shot and SFL settings, exacerbated by the restricted communication between clients. In this paper, we improve the one-shot sequential federated learning for non-IID data by proposing a local model diversity-enhancing strategy. Specifically, to leverage the potential of local model diversity for improving model performance, we introduce a local model pool for each client that comprises diverse models generated during local training, and propose two distance measurements to further enhance the model diversity and mitigate the effect of non-IID data. Consequently, our proposed framework can improve the global model performance while maintaining low communication costs. Extensive experiments demonstrate that our method exhibits superior performance to existing one-shot PFL methods and achieves better accuracy compared with state-of-the-art one-shot SFL methods on both label-skew and domain-shift tasks (e.g., 6%+ accuracy improvement on the CIFAR-10 dataset). Our code and supplementary are available online: https://github.com/NaiboWang/FedELMY.