Neural Architecture Search (NAS) methods have become essential for automating the design of deep neural networks. However, their high computational cost, particularly when training architectures from scratch, remains a significant limitation. This paper proposes an enhancement to the Quantum-Inspired Neural Architecture Search (Q-NAS) algorithm by integrating a dynamically adjustable pre-trained backbone into the search process. The proportion of backbone depth used is jointly optimized with the design of additional building blocks selected from a flexible set of operators. Experiments conducted on four MedMNIST datasets show that the proposed method improves search efficiency, reducing GPU time by up to 66% compared to the baseline. Furthermore, it achieves competitive or superior classification accuracy. Notably, on the PathMNIST dataset, the enhanced QNAS achieved a 3.45% improvement in accuracy over the baseline. These findings demonstrate the effectiveness of combining partial backbone utilization with evolutionary NAS techniques for efficient medical image classification.