I have a project under Git version control that I worked on both a server and my local computer. I originally had the remote origin set as my local computer but I would now
As it seems you have used git clone --depth to clone your local version. This results in a shallow clone. One limitation of such a clone is that you can't push from it into a new repository.
You now have two options:
So, you want to keep your history, eh? This means that you have to unshallow your repository. To do so you will need to add your old remote again.
git remote add old
After that we use git fetch to fetch the remaining history from the old remote (as suggested in this answer).
git fetch --unshallow old
And now you should be able to push into your new remote repository.
Note: After unshallowing your clone you can obviously remove the old remote again.