Existing speech emotion recognition (SER) methods commonly suffer from the lack of high-quality large-scale corpus, partly due to the complex, psychological nature of emotion which makes accurate labeling difficult and time consuming. Recently, transfer learning based methods that exploit the encoders pretrained on large-scale speech corpus (e.g., Wav2Vec2.0 and HuBERT) have shown strong potential for downstream SER tasks. However, task-specific fine-tuning remains necessary for various conversational scenarios of different topics, speakers and languages to achieve satisfactory performance. It generally requires costly encoder retraining for individual SER tasks. To address this issue, we propose to train an adapter with local attributes for efficient transfer learning. Specifically, a weighted average pooling-Transformer (WAP-Transformer) is proposed as a lightweight backbone to enrich the frame-level representation. An adapter with teacher-student branches is exploited for task-agnostic transfer learning, where the student branch is jointly optimized via mask prediction and self-distillation objectives, and the teacher branch is obtained online from the student via exponential moving average (EMA). Meanwhile, local attributes are learned from the teacher branch via unsupervised clustering, which aims to act as a universal model that provides additional semantic-rich supervisions. A statistical attentive pooling (SAP) module is proposed to obtain utterance representation for fine-tuning. To evaluate the effectiveness of the proposed adapter with local attributes, extensive experiments have been conducted on IEMOCAP. Superior performance has been reported, compared to the previous state-of-the-art methods in similar settings.