To define multiple document templates you have to implement multiple content types.
I assume that you want keep the New Document which comes OOTB and additionally your custom document template. If yes, then you should create a separate content type (inherited from Document content type) in which your custom columns should be added and in the
Advanced Settings upload your document template file.
Now in the document library, enable the
Allow management of content types? from the
Advanced Settings in document library.
To manage the order of templates in the New Document option you can specify the order of content type from
Change new button order and default content type link. Please make sure that the default
Document (which will be by default added when you enable the content type for that library) content type is added in the document library.
Hope this helps!!