Alzheimer's disease (AD) is a progressive neurodegenerative disorder that severely affects memory, behavior, and cognitive function. Early and accurate diagnosis is crucial for effective intervention, yet detecting subtle changes in the early stages remains a challenge. In this study, we propose a hybrid deep learning-based multi-class classification system for AD using magnetic resonance imaging (MRI). The proposed approach integrates an improved DeepLabV3+ (IDeepLabV3+) model for lesion segmentation, followed by feature extraction using the LeNet-5 model. A novel feature selection method based on average correlation and error probability is employed to enhance classification efficiency. Finally, an Enhanced ResNext (EResNext) model is used to classify AD into four stages: non-dementia (ND), very mild dementia (VMD), mild dementia (MD), and moderate dementia (MOD). The proposed model achieves an accuracy of 98.12%, demonstrating its superior performance over existing methods. The area under the ROC curve (AUC) further validates its effectiveness, with the highest score of 0.97 for moderate dementia. This study highlights the potential of hybrid deep learning models in improving early AD detection and staging, contributing to more accurate clinical diagnosis and better patient care.