Depending on what host you are using (that is, where your fork is located), there might be some additional internal links, you can't change so easily. For example on Github, the fork is directly linked to the original you forked from. In that case you need to fork the new project again, and work with the new fork.
In that case however you can easily change the URL of the origin repository as well, and just push everything you changed before in your old fork into your new fork.