Reading the thread and having an iPhone4S with IOS6 on it, and wanting to keep IOS6 so the news owner can decide for themselves if they want to upgrade to IOS7 or not, I followed this process:
0) Used the AT&T website form to submit a request to unlock my out-of-contract iPhone4S. After 3-4 days I got a reply email saying my unlock was approved and I was supposed to use the backup and then 'restore iphone' process in iTunes. That, of course, would restore the phone to IOS7, which I do not want to do... so I didn't follow that process, but I had gotten the AT&T unlock approved email.
1) iTunes: Backed up my iphone with iTunes (unencrypted)
2) On iPhone : Settings App -> General -> Reset -> Reset all settings and erase all data
3) Unplugged the iPhone from the computer as the reset was proceeding. Left iTunes running.
4) Went away for a while... so I'm not sure how long the entire reset process took
5) iPhone: shows grey intro screen with a swipe bar at the bottom -what you have with a new iPhone
6) Plugged the iPhone back into the computer (iTunes was still running )
7) iTunes dialog box asking me to upgrade to the newest version of iOS for the device. Options offered: [cancel] [download and update] [download only]. I indicated [cancel]
8) iTunes dialog box indicateing "An update to the carrier settings for your iPhone is available. Would you like to download it now?" I selected [Download and Update]
9) iTunes showed a top level screen saying "Congratulations, your iPhone has been unlocked."
10) as I was not quite ready to sell the phone yet, I now restored the phone from the backup. Which restarts the iPhone after the restore is complete.
11) on reboot of the iPhone, still being plugged into the computer and thus connected to iTunes, iTunes again asked "A new iPhone software version (7.0.4) is available for the iPhone..... Would you like to download it and update your iPhone now?" Again, I chose [cancel]
12) the iPhone then began to sync (apps, music, etc) - this took a LOT longer than the 'restore from backup' part.
13) I got a dialog about some apps not being able to be installed as they are not compatible (presumably iOS7 only versions of apps I had) - clicked OK - the phone was semi-functional as the restore of the additional content was proceeding (had to set the lock code again, for example).
14) Everything seemed restored, including my game states (Tiny Tower for example) and I was still rocking iOS6