Call your local government municipal waste or recycling center. They may accept excess soil from your garden. There may be lawn waste collection days in your neighborhood or dumpsters for this purpose. Keep in mind that many lawn waste areas of your dump or recycling center do not accept soil.
Advertise that you have free garden soil in a local community newsletter, an online site (e.g., Craiglist for your area) or another small local distribution outlet (e.g., Pennysaver). Many gardeners will come pick up your garden soil, especially if it is rich and fertile. If it is just fill dirt, not of much value for a garden, many people need fill dirt for outside landscape or construction projects.
Contact the contractor for your local trash pickup, if applicable. Their name is often on the trashcans and trash trucks. They may provide a service that picks up garden waste, including soil, but probably a fee will be charged.