While machine learning (ML) provides a great tool for image analysis, obtaining accurate fracture segmentation from high-resolution core images is challenging. A major reason is that the segmentation quality of large and detailed objects, such as fractures, is limited by the capacity of the segmentation branch. This challenge can be seen in the Mask Region-based Convolutional Neural Network (Mask R-CNN), which is a common and well-validated instance segmentation model. This study proposes a two-stage segmentation approach using Mask R-CNN to improve fracture segmentation from unwrapped-core images. Two CNN models are used: the first model processes full-size unwrapped-core images to detect and segment fractures; the second model performs a more detailed segmentation by processing smaller regions of the images that include the fractures detected by the first model. In addition, the procedure uses a new architecture of Mask R-CNN with a point-based rendering (PointRend) neural network module that can increase segmentation accuracy. The method is evaluated on approximately 47 m of core from four boreholes and results in an improvement to previous fracture segmentation methods. It achieves an increase in the average intersection over union of approximately 27% from the baseline (one-stage segmentation with standard Mask R-CNN). The enhanced fracture segmentation provides a mean for obtaining an accurate fracture aperture with an average error of less than 1 mm, which represents a reduction of 0.5 mm from the baseline method. This work presents a novel contribution towards developing an ML-based workflow for core-image analysis.