There are several small lakes and parks in the area of Greenville that are good spots for an off-road run. A bit further is Issaqueena Lake, which is a terrific place for a wooded trail run, with occasional lake views. Here are some recommendations:
Oak Grove Lake Park. Located 7 miles east of downtown Greenville. Narrow lake with an 0.8 mile jogging trail around the perimeter with water views.
Legacy Park. Located 7 miles southeast of downtown in the Verdae area. Small park with some trails. Neighboring Verdae is a newish residential area with some quiet roads.
Greer City Park. The town of Greer, located 12 miles northeast of Greenville, has a pleasant park in its middle. Some tree-lined trails, a small pond, and City Hall. About 0.75 miles around. Combine with some of the nice residential streets nearby.
Paris Mountain State Park. Only 6 miles north of Greenville, this is a popular spot for hiking, mountain biking, and trail running. There are more than 15 miles of trails that are great for trail running. Hilly and challenging. Note: trails closed to mountain bikes on Saturdays! TRAIL INFO
Issaqueena Lake Trail. The lake is located in Clemson, 30 miles east of Greenville. A single track, but very runnable trail goes for 2 miles between Issaqueena Lake Rd. and Dam Rd. Mainly wooded, with a couple of lake views. You can also run on Issaqueena Lake Rd., a wooded dirt road with very light traffic, on the north and west sides of the lake.