Is it possible to upgrade Discourse up to a number of commits in the version?

Refer to this topic:

You could try using the version (commit ID) found either on github or in your development page source…

So viewing the source of this page currently:

<meta name="generator" content="Discourse 1.6.0.beta7 - version b624c5cc9403d58d8d45125d7f70e60cddbb9ab9">

I can see the version is b624c5cc9403d58d8d45125d7f70e60cddbb9ab9, this is the ID you could try using in place of tests-passed on your staging / production.

To be clear - I’ve not tested using a commit ID - but it should work.