There are some pleasant parks and ponds in the area around the Market Common shopping mall. There are numerous options for stringing together your own route. A pleasant 1.1 mile path circles the pond at Valor Memorial Garden. Add another ~1 mile around the paths of adjacent Grand Park, which has numerous paths and fields.
For more in this area, there are quiet roads just west of the mall around Horry Technical College and the Woodland Park residential area. You can also run into Myrtle Beach State Park, which has lovely canopied roads and some trails, reaching the State Pier. Good running options to get to the State Park are Mallard Lake Dr., Farrow Pkwy., and an unnamed trail paralleling Pampas Dr. (see Google Maps, bicycling view).