How to hook up old repository history to a new repository that already has commits?

I've moved my repository to another location on my local system, and I used the git init command, which made a new repository and created an entirely new history system. I've made a few commits since that happened. The commits on the old local system do not show up on the current local system, which is to be expected, since I made a completely new repository. The commits only exist on GitHub now, and I can only see them because I put tags on those commits. When I go back to those commits, this shows:

The old commits on the old local system don't show up on the new local environment, because the history was empty, so git reflog does not show the older commits on GitHub.

Git, in the local environment, treats the commits as if they do not exist/are not part of the branch, but I want those commits to be added back onto the tree, so it's easy for me to compare them with each other. (Trying to compare the commits that are not part of the new commit tree shows that they have entirely different commit histories, even though the new commits are changes from the older commits.)

git rebase and git cherry-pick does not work, I've tried those already. git merge does not work, because, in a rush, I've deleted the old local system. The only proof that the commits from the old system exist is the fact that they're on GitHub and have tags attached to them.

In short, I want to add a bunch of commits that are on GitHub to the history of another repository, which already has commits, and push it back onto GitHub so it appears that all those commits are on the same timeline.

I created a new branch at the old commits, merged the branch with the old commits to the current commits and then pushed that back onto GitHub, so it appears that all the commits have one shared history.