Begin by familiarizing yourself with the TVmaze API documentation, particularly the schedule endpoint. The API provides access to TV show schedules in a structured JSON format. Note down the URL for the schedule endpoint, which typically looks like `http://api.tvmaze.com/schedule`.

Ensure you have an AWS account with necessary permissions to create and manage S3 buckets. Go to the AWS Management Console and create a new S3 bucket where you will store the TVmaze schedule data. Make a note of the bucket name and region.

Create a Python script to fetch data from the TVmaze schedule API. Use the `requests` library to make an HTTP GET request to the API endpoint. Parse the JSON response to ensure the data is correctly formatted.

```python
import requests

response = requests.get('http://api.tvmaze.com/schedule')
data = response.json()
```

Process the fetched data as needed. This might include filtering specific fields or reformatting the JSON data to meet your storage requirements. This step is crucial to ensure the data is stored in a way that is useful for future processing or analysis.

Install the AWS SDK for Python, known as `boto3`, to interact with AWS services. Configure your AWS credentials locally using the AWS CLI. Run `aws configure` in your terminal and input your AWS Access Key, Secret Key, region, and output format.

```bash
pip install boto3
aws configure
```

Extend your Python script to upload the processed data to your S3 bucket. Use `boto3` to create a session and upload the JSON data to the specified bucket. Ensure that the data is saved with an appropriate filename, potentially using a timestamp for versioning.

```python
import boto3
import json
from datetime import datetime

s3 = boto3.client('s3')
bucket_name = 'your-bucket-name'
file_name = f'tvmaze_schedule_{datetime.now().strftime("%Y%m%d_%H%M%S")}.json'

s3.put_object(Bucket=bucket_name, Key=file_name, Body=json.dumps(data))
```

To regularly update the data, schedule your script to run at desired intervals using a task scheduler. On Linux, you can use `cron` jobs, or on Windows, you can use Task Scheduler. This will help in keeping your S3 bucket up-to-date with the latest schedule data.

Example of a cron job that runs the script every day at midnight:
```bash
0 0 * * * /usr/bin/python3 /path/to/your/script.py
```

By following these steps, you can efficiently move data from the TVmaze schedule API to an S3 bucket without relying on any third-party connectors or integrations.