public class Destinations
    {
        public List<Data> data = new List<Data>();
    }

    public class Data
    {
        public string Destination { get; set; }
        public string DialCode { get; set; }
    }

you could:

string path = @"D:\tmp3\myconfig.json";

            Destinations ds = new Destinations();

            ds.data.Add(new Data { Destination = "Hereford", DialCode = "91432" });
            ds.data.Add(new Data { Destination = "Hay", DialCode = "01497" });

            //Save
            var j = JsonConvert.SerializeObject(ds);
            System.IO.File.WriteAllText(path,j);

            //Retrieve and add
            var j2 = System.IO.File.ReadAllText(path);
            Destinations ds2 = JsonConvert.DeserializeObject<Destinations>(j2);
            ds2.data.Add(new Data{Destination = "Monmouth",DialCode = "0600"});
            System.IO.File.WriteAllText(path,JsonConvert.SerializeObject(ds2));