Ya, sorry. Apple will only sign the latest software available. If you want the 4S to stay with iOS 6, it has to stay with 6.0. you will never be able to update or restore the 6.0 device unless you do a factory reset on the device itself.
Even if you download the ispw file from a third party source and try using that file to update or restore through iTunes, it always has to go through Apple and it will not be signed because it is not the latest version.
Your only option if you REALLY want the latest iOS 6 firmware is to buy a new or used device with that firmware. eBay is a good spot to check. 6.1.3 is latest for 4S and 6.1.4 (what I am using) is latest for iPhone 5.