Magnetic resonance imaging (MRI) is crucial for diagnosing and monitoring of multiple sclerosis (MS) as it is used to assess lesions in the brain and spinal cord. However, in real-world clinical settings, MRI scans are often acquired with thick slices, limiting their utility for automated quantitative analyses. This work presents a single-image super-resolution (SR) reconstruction framework that leverages SR convolutional neural networks (CNN) to enhance the through-plane resolution of structural MRI in people with MS (PwMS). Our strategy involves the supervised fine-tuning of CNN architectures, guided by a content loss function that promotes perceptual quality, as well as reconstruction accuracy, to recover high-level image features. Extensive evaluation with MRI data of PwMS shows that our SR strategy leads to more accurate MRI reconstructions than competing methods. Furthermore, it improves lesion segmentation on low-resolution MRI, approaching the performance achievable with high-resolution images. Results demonstrate the potential of our SR framework to facilitate the use of low-resolution retrospective MRI from real-world clinical settings to investigate quantitative image-based biomarkers of MS.