Start by configuring a Jenkins job that will export the data you need. This can be done by creating a job that runs a script or command to generate the output in a structured format such as CSV or JSON. Ensure the data is saved to a location accessible by your system.

After the build process, use Jenkins post-build actions to archive the data output. You can use the "Archive the artifacts" option to ensure the data files are stored in a location where they can be accessed or downloaded later.

On the machine where you wish to run the script, install the Google Sheets API Python client. You can do this by running:
```bash
pip install --upgrade google-api-python-client google-auth-httplib2 google-auth-oauthlib
```

Create a project in Google Cloud Platform and enable the Google Sheets API. Navigate to the "Credentials" section to create an OAuth 2.0 client ID. Download the JSON file containing the credentials and save it securely on the machine.

Develop a Python script that reads the exported data file from Jenkins and uploads it to Google Sheets. Use the Google Sheets API to authenticate using the credentials JSON file and append the data to a specific Google Sheets document. Here is a simplified example snippet:
```python
from google.oauth2.service_account import Credentials
from googleapiclient.discovery import build
import csv

# Authenticate and construct service
creds = Credentials.from_service_account_file('path/to/credentials.json')
service = build('sheets', 'v4', credentials=creds)

# ID of the Google Sheets document
SPREADSHEET_ID = 'your-spreadsheet-id'
RANGE_NAME = 'Sheet1!A1'

# Read CSV data
with open('path/to/data.csv', newline='') as csvfile:
reader = csv.reader(csvfile)
data = list(reader)

# Prepare the data for uploading
body = {
'values': data
}

# Upload the data
service.spreadsheets().values().update(
spreadsheetId=SPREADSHEET_ID,
range=RANGE_NAME,
valueInputOption='RAW',
body=body
).execute()
```

Incorporate the Python script into your Jenkins job. You can add a build step that runs the Python script after the data export step. This ensures that the data is automatically pushed to Google Sheets after each Jenkins build.

Once the data is uploaded, verify its integrity by checking the Google Sheets document. Implement basic error handling in your Python script to catch exceptions and log errors during the upload process. This will help in troubleshooting if the data fails to upload.

By following these steps, you can efficiently move data from Jenkins to Google Sheets using a direct approach without relying on third-party connectors or integrations.