Improved X-ray sources and optics now enable X-ray imaging resolution down to approximately 50 nm for laboratory-based X-ray microscopy systems. This offers the potential for submicrometre resolution in tomography; however, achieving this resolution presents challenges due to system stability. We describe the use of software methods to enable submicrometre resolution of approximately 560 nm. This is a very high resolution for a modest laboratory-based point-projection X-ray tomography system. The hardware is based on a scanning electron microscope, and benefits from inline X-ray phase contrast to improve visibility of fine features. Improving the resolution achievable with the system enables it to be used to address a greater range of samples.