This is the "Next Tab" Firefox extension
- https://addons.mozilla.org/en-US/firefox/addon/next-tab/ -
with some of my own minor improvements on top.

It basically lets you open a page in a tab next to the current tab
either via a context menu (right click > open the link next to current tab)
or via the file menu item to launch a new empty tab next to the current one.
The latter can be quickly done via keyboard with Alt+F > X.

Firefox now opens related tabs together by default, but if you're like me
and prefer to have the old (new tab at end) behavior by default, and manually
control which tabs go next to the current tab, you can go to about:config
and set browser.tabs.insertRelatedAfterCurrent to false, then install
this extension.