Uncognito por BreadFish64

Usage

Ensure that "Run in private browsing" is enabled in about:addons.

Desktop

Click the extension button in the browser toolbar. The extension will attempt to reopen the URL in an existing private window, otherwise it will create a new window.

Mobile

Click the extension button at ⋮ > Extensions > Uncognito. This currently create a non-private tab which still appears in the private section of the tab switcher. Toggling the tab back to private is currently not supported since I have not found a way to create a new incognito tab using the APIs available on Android.