New Tab Context de Luke Baker

An intentionally minimalist add-on with a solitary purpose: to add an item to the tab context menu that opens an adjacent new tab. It doesn't do anything else, it doesn't do it differently, nor will it ever.

Tweaks
The menu entry has an icon and it ends up at the bottom. It seems webextensions can't change that. But you can, by adding the following to your userChrome.css file and restarting Firefox afterwards:
/* New Tab Context - move the menu item to the top */
#tabContextMenu menuitem[label="New Tab"] { -moz-box-ordinal-group: 0 !important; }
/* New Tab Context - hide the icon */
#tabContextMenu menuitem[label="New Tab"] .menu-iconic-icon { display: none !important; }

Credits
Icon by Alessandro Rei.

Alternatives
If you're also interested in keyboard shortcuts and duplicating tabs, have a look at New Tab Beside.