Prior representative ReAct-style approaches in autonomous Software Engineering (SWE) typically lack the explicit System-2 reasoning required for deep analysis and handling complex edge cases. While recent reasoning models demonstrate the potential of extended Chain-of-Thought (CoT), applying them to the multi-turn SWE task creates a fundamental dilemma: retaining full reasoning history leads to context explosion and `Lost-in-the-Middle'' degradation, while discarding it would force the agent to redundantly re-reason at every step. To address these challenges, we propose SWE-AGILE, a novel software agent framework designed to bridge the gap between reasoning depth, efficiency, and context constraints. SWE-AGILE introduces a Dynamic Reasoning Context strategy, maintaining a `sliding window'' of detailed reasoning for immediate continuity to prevent redundant re-analyzing, while compressing historical reasoning content into concise Reasoning Digests. Empirically, SWE-AGILE sets a new standard for 7B-8B models on SWE-Bench-Verified using only 2.2k trajectories and 896 tasks. Code is available at https://github.com/KDEGroup/SWE-AGILE.