The only way I see it, you should write your own converter. But half of work is already done in class StringEnumConverter. We can override only ReadJson method

class Program
{
    static void Main(string[] args)
    {
        const string json = @"{
                'Name': 'abc',
                'Type':'Type4'
            }";

        // uncomment this if you want to use default value other then default enum first value
        //var settings = new JsonSerializerSettings();
        //settings.Converters.Add(new FooTypeEnumConverter { DefaultValue = FooType.Type3 });

        //var x = JsonConvert.DeserializeObject<Foo>(json, settings);

        var x = JsonConvert.DeserializeObject<Foo>(json);
    }
}

public class Foo
{
    public string Name { get; set; }

    public FooType Type { get; set; }
}

public enum FooType
{
    Type1,
    Type2,
    Type3
}

public class FooTypeEnumConverter : StringEnumConverter
{
    public FooType DefaultValue { get; set; }

    public override object ReadJson(JsonReader reader, Type objectType, object existingValue, JsonSerializer serializer)
    {
        try
        {
            return base.ReadJson(reader, objectType, existingValue, serializer);
        }
        catch (JsonSerializationException)
        {
            return DefaultValue;
        }
    }
}

Let's say we have the following json string:

[
    {
        "Name": "abc",
        "MyEnumValue": "Type1"
    },
    {
        "Name": "abcd",
        "MyEnumValue": "Type2"
    },
    {
        "Name": "abcde",
        "MyEnumValue": "Type3"
    }    ,
    {
        "Name": "abcdef",
        "MyEnumValue": "Type4"
    }
]

and the following class and enum:

public class MyClass
{
    public string Name { get; set; }

    public MyEnum MyEnumValue { get; set; }
}

public enum MyEnum
{
    Type1,
    Type2,
    Type3
}

As it can be noticed, the json string array contains item (the last one), that cannot be correctly mapped to the MyEnum. To avoid deserialization errors you can use the following code snippet:

static void Main(string[] args)
{         
    var serializationSettings = new JsonSerializerSettings
    {
        Error = HandleDeserializationError
    };

    var lst = JsonConvert.DeserializeObject<List<MyClass>>(jsonStr, serializationSettings);
}

public static void HandleDeserializationError(object sender, ErrorEventArgs errorArgs)
{
    errorArgs.ErrorContext.Handled = true;
    var currentObj = errorArgs.CurrentObject as MyClass;

    if (currentObj == null) return;
    currentObj.MyEnumValue = MyEnum.Type2;            
}

where the jsonStr variable is the posted json string above. In the above code sample, if MyEnumValue cannot be correctly interpreted, it is set to a default value of Type2.

Example: https://dotnetfiddle.net/WKd2Lt

An alternative, if you don't want to create a custom converter, is to map it to a private string field in your DTO and then use Enum.TryParse in that field's property getter:

public class MyClass
{
    [JsonProperty("MyEnumValue")]
    private string myEnumValue;

    public string Name { get; set; }

    [JsonIgnore]
    public MyEnum MyEnumValue 
    { 
        get
        {
            MyEnum outputValue = MyEnum.Default;
            Enum.TryParse(myEnumValue, out outputValue);
            return outputValue;
        }
    }
}