In recent years, with large language models (LLMs) achieving state-of-the-art performance in context understanding, increasing efforts have been dedicated to developing LLM-enhanced sequential recommendation (SR) methods. Considering that most existing LLMs are not specifically optimized for recommendation tasks, adapting them for SR becomes a critical step in LLM-enhanced SR methods. Though numerous adaptation methods have been developed, it still remains a significant challenge to adapt LLMs for SR both efficiently and effectively. To address this challenge, in this paper, we introduce a novel side sequential network adaptation method, denoted as SSNA, for LLM enhanced SR. SSNA features three key designs to allow both efficient and effective LLM adaptation. First, SSNA learns adapters separate from LLMs, while fixing all the pre-trained parameters within LLMs to allow efficient adaptation. In addition, SSNA adapts the top-a layers of LLMs jointly, and integrates adapters sequentially for enhanced effectiveness (i.e., recommendation performance). We compare SSNA against five state-of-the-art baseline methods on five benchmark datasets using three LLMs. The experimental results demonstrate that SSNA significantly outperforms all the baseline methods in terms of recommendation performance, and achieves substantial improvement over the best-performing baseline methods at both run-time and memory efficiency during training. Our analysis shows the effectiveness of integrating adapters in a sequential manner. Our parameter study demonstrates the effectiveness of jointly adapting the top-a layers of LLMs.