Without writing a custom macro to do this, I would suggest this: Make the changes to your doc and save it. Before closing the doc, perform a "Save as,"
navigate to the template and save it as the template. You will get a pop-up asking if you wish to replace the existing one; say yes.

If you should get a pop-up telling you that you cannot save as a document already open (i.e., the template, which Word may consider to be in use, since a document based on it is open), tings get a bit more cumbersome. You would have to alter the template name temporarily, in the "Save as" box, perhaps just adding a 1 to the end (for instance, mytemplate1.dot). After saving this, and then closing your document, you would then have to delete or archive your original template and rename the new one without the 1.