Is it possible to add an Instagram post to a project after publishing it from the network?

Yes, this is certainly possible. It used to be that you could not make Instagram posts on Media Injection, but this has been changed and now it is possible to create Instagram posts straight from the Media Injection Calender. 

However, in the case that you already have something created within Instagram itself you can still retrieve the information and add it to certain projects.

To do so, go to the 'Project messages' stream and find the message you wish to add to a project and click on the grayed out suitcase symbol. Click the suitcase and select the project you wish to add the post to underneath the post. 

Please note that it is only possible to add Instagram messages to projects that have Instagram added to their list of linked channels.

Additional information: Note that this isn't just an option for Instagram. Other posts that are published via the channel itself or Facebook ads can be added to a project later on as well.