I recently needed to upgrade my application, the Advanced Geocaching Tool, because a website change at geocaching.com rendered the main function of the tool, downloading geocaches, dysfunctional. The package is currently in extras-testing, and awaits the ten votes as well as the 10 days until promotion to extras.

As this affects all users of the app, I wonder if there is some way to bypass the 10 day rule for this type of minor updates. This may happen any time again, and leave the users, which don't have extras-devel installed, out in the rain.

AFAIK no, as even minor updates could lead to problems (and that's the whole point of the Extras QA process). It's only ten days, which is not too long, and you can do instant promotions from Extras-Devel to Extras-Testing, of course. Also, PR1.2's Application Manager allows single .debs to be installed from the file manager, so you can instruct your users to download only the updated .deb from Extras-Testing (without even enabling the repository) in the mean time (again, only one week and three days).

You can tell your users to vote for the package in Extras-Testing after testing the package, so you get your 10 thumbs up in the 10 days and can promote right after that. Another helpful hint is to promote packages you want to get into Extras very early to Extras-Testing, as the 10 days start to count from the time the package is put into Extras-Testing.