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A simple extension that opens new tab or duplicates right beside the current tab. Many other browsers have "open a new tab right next to current tab" feature built in but not Firefox, and so this extension was created.

Instructions

1. Access context menu by right-clicking on a tab or anywhere on a webpage.
2. Hotkeys:
   
   • New Tab (Ctrl/Command+Shift+1)
   • Duplicate Tab (Ctrl/Command+Shift+2)

Note
Due to Firefox's new extension development standards, developers can no longer modify context menu items' ordering. We know this is mildly annoying, but there's always shortcut keys.
UPDATE: This extension will continue to work even with Firefox's new Duplicate Tab function starts rolling out as Firefox's implementation d̶o̶e̶s̶ ̶n̶o̶t̶ ̶d̶u̶p̶l̶i̶c̶a̶t̶e̶ ̶t̶a̶b̶ ̶b̶e̶s̶i̶d̶e̶ ̶t̶h̶e̶ ̶c̶u̶r̶r̶e̶n̶t̶ ̶t̶a̶b̶ does not duplicate beside any tabs being right clicked on.
UPDATE v.2.2: We hear ya. New Tab function now directly accessible upon right-clicking a tab, without the hassle of having to reach a sub-menu. The Duplicate Tab function will still work by using hotkey, as Firefox has implemented this feature for some time anyway.

This extension has not been (and will not be) tested in Firefox Developer Edition (or any other experimental releases) as it was not our intention to develop this outside of Firefox's mainstream releases. Thank you for your understanding and for using our extension.

Fun Fact
The extension author actually uses this extension on a daily basis!