Local authorities are likely to have data they could publish as open data provide but do make sure there there are no legal restrictions attached to it before adding it. Authorities such as Hampshire County Council are already releasing large datasets, such as Rights of Way information. You can find out more about your local authority's progress in opening up data at Openly Local. The Open Government Licence can be used by public sector bodies when they release data.
See if you can involve your local bus company to provide traces of their routes to improve coverage of public transport. Local walking or cycling groups may be able to help to map out missing Rights of Way in your area. Schools are worth contacting too as improving mapping of the local area could be a good project for geography lessons.