Purpose: The scarcity of high-quality curated labeled medical training data remains one of the major limitations in applying artificial intelligence (AI) systems to breast cancer diagnosis. Deep models for mammogram analysis and mass (or micro-calcification) detection require training with a large volume of labeled images, which are often expensive and time-consuming to collect. To reduce this challenge, we proposed a novel method that leverages self-supervised learning (SSL) and a deep hybrid model, named \textbf{HybMNet}, which combines local self-attention and fine-grained feature extraction to enhance breast cancer detection on screening mammograms. Approach: Our method employs a two-stage learning process: (1) SSL Pretraining: We utilize EsViT, a SSL technique, to pretrain a Swin Transformer (Swin-T) using a limited set of mammograms. The pretrained Swin-T then serves as the backbone for the downstream task. (2) Downstream Training: The proposed HybMNet combines the Swin-T backbone with a CNN-based network and a novel fusion strategy. The Swin-T employs local self-attention to identify informative patch regions from the high-resolution mammogram, while the CNN-based network extracts fine-grained local features from the selected patches. A fusion module then integrates global and local information from both networks to generate robust predictions. The HybMNet is trained end-to-end, with the loss function combining the outputs of the Swin-T and CNN modules to optimize feature extraction and classification performance. Results: The proposed method was evaluated for its ability to detect breast cancer by distinguishing between benign (normal) and malignant mammograms. Leveraging SSL pretraining and the HybMNet model, it achieved AUC of 0.864 (95% CI: 0.852, 0.875) on the CMMD dataset and 0.889 (95% CI: 0.875, 0.903) on the INbreast dataset, highlighting its effectiveness.