Accurate classification of brain tumors is crucial for effective treatment planning and patient management. This study presents a new hybrid deep learning classification method based on transfer learning by feature extraction to automate the categorization of MRI brain image datasets into four classes: meningioma, glioma, pituitary tumor, and no tumor. The proposed method combines a finely-tuned ResNet-50 model, a state-of-the-art convolutional neural network architecture, with optimized Softmax Regression (SR) for classification. The study explores the use of data augmentation techniques and evaluates the model's performance on both augmented and unaugmented images. The results demonstrate that the proposed method achieves an impressive accuracy of 98.4%, outperforming existing methods for automatic brain tumor detection. Furthermore, a detailed comparative analysis is presented to evaluate the proposed model's accuracy and efficiency against previous state-of-the-art hybrid models for brain tumor classification. The study suggests that the proposed methodology could be employed as a diagnostic tool to aid radiologists in identifying questionable brain regions, potentially improving the accuracy and efficiency of brain tumor diagnosis.