Below is the procedure to jailbreak iOS 6.1.3 using the latest available jailbreak redsn0w tool for iPhone 3gs, iPod Touch 4G & iPad. Before you start be careful and note that this is a tethered jailbreak which means you will need to use redsn0w for boot tethering if you need to reboot your device.
Steps On How To Jailbreak iOS 6.1.3 Semi Untethered
Download redsn0w 0.9.15b3 from this page.
Download the iOS 6.0 Firmware for your device iPhone, iPod Touch or iPad
Run the redsn0w.exe file as administrator and connect your device to computer. If you face errors running the redsn0w you may need to set the compatibility mode to Windows XP Service Pack 2 and open the redsn0w executable file by right clicking and selecting the option of Run as administrator.
Go to “Extras” => “Select IPSW” then select the 6.0 firmware file which you already downloaded and click Open. No need to confuse we are surely using 6.0 to jailbreak and 6.1.3 is already on your device. This is necessary step to jailbreak iOS 6.1.3 otherwise you will face unexpected jailbreak errors.
Go back to main redsn0w screen and click “Jailbreak”. Click continue and ensure “Install Cydia” is checked.
Now you shall be prompted to turn off your device and enter DFU mode as per the instructions on the screen. Just follow the steps and your device will go DFU mode and you will see activity on redsn0w and your device screen. After this steps are completed now your iPhone, iPod or iPad is tethered jailbroken and needs to boot tethered.
As I told in the beginning this jailbreak is Semi Untethered therefore you will need to complete below steps every time your device powers down completely or restarts. You will also have to do these steps otherwise Cydia & Safari will not open.
Boot Tethered For iOS 6.1.3 Jailbreak
Open Redsn0w and got to ”Extras” => “Select IPSW” => Select 6.0 ipsw file.
Go back to main screen and Select “Just boot” and then follow the process until device becomes ready after reboot.
Thats all your iPhone 3GS, iPod Touch 4G or iPad is now jailbroken on 6.1.3.