When you upload an image into the media library, WordPress creates a new post of a special type. The title tag is stored as a post title, the caption as excerpt, the alt tag as a special field. So, if you call the image through the library it will know all the metadata. For instance, if you use it as a featured image, the title and alt tags will be filled in automatically.
However, if you call the image directly from a post by including the URL, you are bypassing the media library. This leaves you entirely free to include new alt and title tags.
In fact, if you insert an image in a post through the media library in the code edit mode (as opposed to visual editor), you can edit the tags without affecting the metadata saved in the media library. So that will not affect your old post.