Now you can jailbreak iOS 6.1.3 running with the latest software version Redsn0w for Windows and Mac here below.
How to Jailbreak iOS 6.1.3 with Redsn0w 0.9.15b3
Step 1: Download iOS 6.1.3 IPSW and firmware file for iOS 6.0.
Step 2: Download the latest version Redsn0w 0.9.15b3 here (Step 2).
Step 3: Run Redsn0w 0.9.15b3 in Administrator mode if you’re on Windows or make a click, hold down Ctrl, and select “Open” from the new menu, if you are a user OS X Mountain Lion.
Step 4: Go to the menu Extras => Select IPSW and specify in the IPSW-file iOS 6.0 (not iOS 6.1.3) on the device for which you are performing the jailbreak.
Step 5: Go back to the first page and click on the Jailbreak button, then check Install Cydia and enter DFU mode following on screen instructions.
Step 6: Now Redsn0w will perform the iOS 6.1.3 jailbreaking. After completion clicking “Autoboot this device when it connects in DFU mode” to switch to tethered jailbreak. After the reboot you will get a device running jailbroken iOS 6.1.3 with Redsn0w.
From now on every reboot or turn off the iPhone 4 / 3GS or iPod touch 4G you will have to connect the device to your computer and turn it into a jailbreak with a special function in Redsn0w. To do this, just go to the section Extras and click the button Just boot.
