Begin by accessing the Mailjet SMS API. Sign up for a Mailjet account if you haven't already, and navigate to the API documentation to understand the endpoints available for SMS data retrieval. Generate your API key and secret, which will be used to authenticate your requests.

Write a script using a programming language like Python to fetch the SMS data. Use the `requests` library to send a GET request to the appropriate Mailjet SMS API endpoint. For example, use:
```python
import requests

api_key = 'your_api_key'
api_secret = 'your_api_secret'
url = 'https://api.mailjet.com/v4/sms'

response = requests.get(url, auth=(api_key, api_secret))
sms_data = response.json()
```
This will return the SMS data in JSON format.

Process the JSON data obtained from the API response. Extract the necessary fields you want to move to MySQL (e.g., message content, recipient, status, timestamp). Iterate through the JSON objects to structure the data for database insertion.

Ensure you have a MySQL server running and set up a database and table to store the SMS data. Use the MySQL command line or a tool like phpMyAdmin to create a database, and then create a table with columns matching the extracted data fields.

```sql
CREATE DATABASE sms_data;
USE sms_data;
CREATE TABLE sms_messages (
id INT AUTO_INCREMENT PRIMARY KEY,
message_content TEXT,
recipient VARCHAR(255),
status VARCHAR(50),
timestamp DATETIME
);
```

Use a MySQL connector library in your script to establish a connection to the MySQL database. In Python, you can use `mysql-connector-python`:
```python
import mysql.connector

connection = mysql.connector.connect(
host='localhost',
user='your_username',
password='your_password',
database='sms_data'
)
cursor = connection.cursor()
```

Loop through the parsed SMS data and insert each record into the MySQL table using an `INSERT` statement. Ensure to handle exceptions and commit the transaction to save the changes.

```python
for sms in sms_data['messages']:
insert_query = """
INSERT INTO sms_messages (message_content, recipient, status, timestamp)
VALUES (%s, %s, %s, %s)
"""
data_tuple = (sms['text'], sms['to'], sms['status'], sms['date_created'])
cursor.execute(insert_query, data_tuple)

connection.commit()
```

After inserting all data, close the database connection and handle any potential errors gracefully. Ensure that your script logs any errors for debugging.

```python
cursor.close()
connection.close()
```

By following these steps, you can efficiently transfer data from Mailjet SMS to a MySQL database without using third-party connectors or integrations.