Capture flask request when entering and leaving the application

I'm struggling a bit with writing decorators to capture and modify requests within my flask application once the requests enters the application context and leaves the application context.

My function in a flask application looks like below. I want the control of requests at two points to add custom headers and params:

1. before the request goes inside get_foo function and,
2. just before requests.get('http://third-party-service/v1/api') so I can attach any custom header or query param just before this request leaves the application.

@app.route('/my/v1/api')
def get_foo():
   ...
   ...
   r = requests.get('http://third-party-service/v1/api')
   ...
   ...
   return 'success!'

The former part I'm able to do using @app.before_request decorator where I get control of the request once it reaches the get_foo function.

@app.before_request
def before_request_callback():
    method = request.method
    path = request.path

    print(method, path)

The latter part I'm not able to do. If I use @app.after_request decorator, I get the control once the request has been process and I have got the response from http://third-party-service.

I debugged sentry sdk and they're able to take control of the requests once the request leaves the application. I tried out to follow how they have implemented it by following their piece of code from the github repo (https://github.com/getsentry/sentry-python) but wasn't able to do so hence posting the question over here.

from Capture flask request when entering and leaving the application