Here is a way to quickly update a Tails ISO (or IMG) image file

... by patching the image file you already have, avoiding downloading the whole new version.

Such a 'cold' (not live) upgrade also protects against some types of security breaches.

Pre-requisites

You must already have an image file of some previous version of Tails.

You must also have or obtain right now:

an executable xdelta3
an executable GPG (GnuPG)
the official OpenPGP public signing key stored in your GPG keyring, in order to check the new image file (it MUST BE identical to the official one published by the Tails project). To obtain this public key: download it then invoke gpg --import tails-signing.key .
If you don't know anything about all this nor want to explore then the present document is not for you, therefore upgrade Tails using the standard procedure. If you don't know and want to explore you may first install Tails from Debian or Ubuntu using the command line and GnuPG (then be prepared to benefit from this document).

Process (bird's view)

This section describes the process. The "Usage" section, below, contains detailed instructions.

You will:

download a VCDIFF patchfile, created by me (using the 'xdelta3' software), which describes the differences between the old version of the Tails image file (you already have) and the new one.
For example a file named 'tails-amd64-6.15.1.iso_to_6.16.xdelta3' enables you to patch the Tails AMD 6.15.1 ISO image file into a 6.16 version. It is useless if you don't already have the file named 'tails-amd64-6.15.1.iso'.
invoke 'xdelta' which will apply the patch, creating the new ISO file
check this new image file (using GPG)
use it to boot Tails

Usage

To update an image file:

stop Tails (because during this process you will modify the image file it uses!)
on a command line (the directory containing your Tails image file must be the current directory), invoke the 6 following commands:
1. export UPDTAILS_IMAGE_TYPE=iso # replace 'iso' with 'img' if you want to update an IMG file
2. export Tails_PREVIOUS_VERSION=6.15.1 # replace '6.15.1' by the version number of the image file you already have
3. export Tails_VERSION=6.16 # replace '6.16' by the version number of the newest published version
4. wget --no-clobber http://makarevitch.com/Tails/upd_xdelta/tails-amd64-${Tails_PREVIOUS_VERSION}.${UPDTAILS_IMAGE_TYPE}_to_${Tails_VERSION}.xdelta3 # download the pertinent patchfile. If the file doesn't exist then there is no available patchfile for you, sorry.
5. wget --no-clobber https://tails.net/torrents/files/tails-amd64-${Tails_VERSION}.${UPDTAILS_IMAGE_TYPE}.sig # download the official GPG seal
6. xdelta3 -d -stails-amd64-${Tails_PREVIOUS_VERSION}.${UPDTAILS_IMAGE_TYPE} tails-amd64-${Tails_PREVIOUS_VERSION}.${UPDTAILS_IMAGE_TYPE}_to_${Tails_VERSION}.xdelta3 tails-amd64-${Tails_VERSION}.${UPDTAILS_IMAGE_TYPE} # apply the patchfile
7. gpg --verify tails-amd64-${Tails_VERSION}.${UPDTAILS_IMAGE_TYPE}.sig # check the new image file
If it does NOT produce a gpg: Good signature from "Tails developers (offline long-term identity key)" message then something failed (it never happened to me nor to numerous friends) and DO NOT USE the new image file.

img to iso, and iso to img

If you already have the 'img' file and want the 'iso' file there is an adequate patchfile (for example named 'tails-amd64-6.16.img_to_6.16.iso.xdelta3' .

To transmute an 'iso' file into the corresponding 'img' file use 'tails-amd64-6.16.iso_to_6.16.img.xdelta3' .