I'm not a tennis player, but my husband said that most of the city/county parks that have courts, also have a backboard. If there is a specific area you're interested in, you might try calling
Greenville County Recreation District at (864)288-6470. Here's a link to City of Greenville Park facilities:
Greenville City Parks & Rec (http://www.greenvillesc.gov/city_services/facilities_locations_amenities.asp - broken link)