Initiate Partner Black Tab login without using Subscriber Support tab

Afraid this isn't going to be possible. When you view a subscriber, a separate session is launched on the subscriber org's instance so you can view data about that org. The confirm token initiates that request. The tokens change, so there is nothing you can hardcode, derive to construct the URL.

There's a button available on the License object, we just need to update the License page layout and add the button "Login into Subscriber Console" to layout.

This button exactly does what we looking for, it opens the organisation details page (the same page that we open by clicking on Org name in Subscriber results/rows).

[Update:9 Oct 2019] You can further extend the button into a custom field and made it show as a link, and then add that field in Search Layouts so you can click on the same (from search results or related list) and directly login to customer org.