I mean the only way to send data out is through the POST method ? (leaving out CONNECT)
No. You can encode data in the URL as well. And maybe it doesn't even use HTTP. You can also encode data in Cookies that are sent through a GET request. The idea that only POST is able to send data is a common misconception. HTML forms aren't even restricted to use POST; you can specify a method.
If it uses HTTP then your best shot is to look for HTTP requests that you didn't make. Then scan through this list and check them because there's probably a lot of background software/services that do all sorts of HTTP requests as well (such as checking for available updates and whatnot).