link to add a document in doc library

I need to create a custom web part which has a clickable link only.

Expected behaviour : Every doc lib has "Add a document", if you click that file dialog it will open up a file upload pop-up, file uploads & my custom application page opens up which will eventually used to fill up multiple custom columns my doc lib has. 

I need to implement that "Add a document" link behavior in my custom web part (WP), so that I can use that WP througout the site & be able to upload documents without going to doc lib, simple.

I got the upload form link using fiddler but that is opening file upload dialog in new tab & does not go to next page, where I can fill up other custom column values.

For example link : 

<a href="/sites/Contracts/_layouts/15/Upload.aspx?List=%7b89225CDC-B61A-45FC-8DC3-AC7DD97578CA%7d&amp;RootFolder=&amp;IsDlg=1"> Upload a doc</a>

Above one takes me to new tab & opens small "file upload" dialog in full page. Upon add a doc, it does not lead anywhere.

I am allowed to code server-side or client site.