Is this thread still active if so, allow me to say something...
I got my iPhone 4S as a gift for my birthday. When i switched it on, i was devastated and tried to find a solution to downgrade to ios 6.1.3 in hopes of making my sluggish iphone 4s faster. I downloaded an ipsw file and tried to restore it to 6.1. but no avail. I lost all my data and was very dissappointed. I decided not to do it ever and ever again. After a few months, i decided to try again once more as i was more familiar on the procedure to downgrade. But when i checked the ipsw website, i was COMPLETELY shocked. I sent a suggestion to Apple for Apple to start signing 6.1.3 again. I never heard back since.
Is there any other method to downgrade? PLEASE share me ur thoughts and ideas.
Apple usually doesn't sign previous iOS versions. Once they are unsigned, there is no way to restore them to your iOS device using iTunes. 6.1.3 was an exception for the longest time, it was actually signed for years for the 4s and iPad 2, but that's also unsigned now.
However there is a method, but it requires you to jailbreak your device. You can use the CoolBooter tweak to install a second partition (any supported one for your device).
Install 9.3.6 (latest version) on your 4s, if you don't already have it. Then jailbreak your iPhone 4s. Open Cydia, and search for the CoolBooter tweak and install it. After that, open CoolBooter and select your desired firmware (like 6.1.3). Now you can reboot your device with CoolBooter and it will boot into iOS 6.1.3. It's a new partition so there will be no files / music / pictures etc... If you turn it off, it will boot into iOS 9.3.6 again, meaning that you eventually have to jailbreak again first (unless it's an untethered one) so you can open up your CoolBooter tweak again. After that, you can boot into iOS 6 again. So i recommend you to just leave your device powered on (just leave it at the charger when you don't use it) while in iOS 6.
It's quite complicated but it works. See YouTube for tutorials.
See: https://coolbooter.com