A way to limit the time frame in between the pushes to GitHub

Question

I have a GitHub action being triggered on a push to a certain protected branch. I want to prevent behavior when we have concurrent running GitHub actions if I was to push to this branch several times in a small time frame.

Now I know there is a concurrency parameter in GitHub actions that I can use to queue an action until the previous one is completed. However, since I also have a git push within the GitHub action (and the action is set up not to be triggered by that particular push), it creates a lot of issues where what ends up happening is that the only workflows to actually go through are the first one (the main push) and the internal push within the workflow. So, the other pushes would just be cancelled by the laws of the concurrency parameter. So I was wondering if someone has a way around this.