Information leaks in general could take any form. Any channel through which you can communicate with the server could leak information. Just a random example: imagine a pdf generation service that takes a doc file and creates a pdf. You may provide a bogus doc file that makes the service dump a chunk of heap or stack memory into the generated pdf, which you would then download, and then provide a second doc that completes the exploit.
In general, how you will go from step one to step two is heavily dependent on the target system and the kind of vulnerabilities being exploited. In many cases there will be way more than just two steps, so automation is not a bad idea. If automation is indeed an option, you may want to create a script that performs the steps in sequence. Back to the web server example, you could write some Python or Ruby script that sends the first request, reads the response, extracts relevant information, prepares the second request computing addresses where appropriate, and sends the final request to the server to finish exploitation.