<script type="text/javascript" src="https://code.jquery.com/jquery-1.12.4.js"></script>
    <script type="text/javascript">
        function GetFieldOptions() {
            return $.ajax({
                url: _spPageContextInfo.webAbsoluteUrl + "/_api/web/lists/GetByTitle('mylist')/Fields(guid'2e257823-b8c6-411d-80f3-9f2cc913be0f')",
                type: "GET",                
                headers: {
                    "accept": "application/json;odata=verbose",                    
                    "Content-Type": "application/json;odata=verbose"
                }
            })
        }
        function updateChoiceField() {
            GetFieldOptions().done(function (data) {
                var FieldOptions = data.d.Choices.results;
                FieldOptions.push("NewOption");
                $.ajax({
                    url: _spPageContextInfo.webAbsoluteUrl + "/_api/web/lists/GetByTitle('mylist')/Fields(guid'2e257823-b8c6-411d-80f3-9f2cc913be0f')",
                    type: "POST",
                    data: JSON.stringify({ "__metadata": { "type": "SP.FieldChoice" }, "Choices": { "__metadata": { "type": "Collection(Edm.String)" }, "results": FieldOptions } }),

                    headers: {
                        "accept": "application/json;odata=verbose",
                        "X-RequestDigest": $('#__REQUESTDIGEST').val(),
                        "Content-Type": "application/json;odata=verbose",
                        "X-HTTP-Method": "PATCH",
                        "IF-MATCH": "*",
                    },
                    success: function (data) {
                        console.log(data);
                    },
                    error: function (error) {
                        alert(JSON.stringify(error));
                    }
                });
            })

        }
    </script>
    <input id="Button1" onclick="updateChoiceField()" type="button" value="updateChoiceField" />