iOS 6.1.3 was released last Spring, along with fixes that closed iOS 6.1.3 jailbreak. If you haven’t upgraded your firmware, don’t get it done since you will lose the jailbreak. For those who have up-to-date device to a more recent 6.1.3 you can't downgrade and re-jailbreak using evasi0n.  However, for those who have an A4 or older device you are able to jailbreak iPhone on iOS 6.1.3 using Redsn0w. But remember that this jailbreak is tethered. It's untethered only for iPhone 3GS with the old bootrom. So, for anybody with a pre A5 device, here’s how you can jailbreak iOS 6.1.3.
Supported Devices
  • iPhone 4,
  • iPod Touch 4
  • iPhone 3GS
Make certain your system is running on iOS 6.1.3 before you begin. Also there are some other requirements:

Jailbreak iPhone 4, 3GS Redsn0w for iOS 6.1.3 | Guide

Step 1 Open Redsn0w, choose Extra then Select IPSW.
Step 2 Direct Redsn0w to iOS 6 IPSW you downloaded earlier. Step 3 Now, in the primary screen in Redsn0w choose Jailbreak and make certain you do as instructed which will appear on screen. Step 4 When you're requested to, select Install Cydia. Step 5 Next you'll be requested to place device into DFU mode - Redsn0w provides you with full instructions regarding how to do that. Step 6 Redsn0w should now start the tethered jailbreak process on your device. This will take a maximum of a few moments to accomplish. You will notice codes running on screen of the phone. Step 7 After the process will be completed you will need to boot your device tethered and I will show you how to do that. Boot Tethered Remember this jailbreak is tethered. When your device reboots or else you lose battery, you'll be stuck on Apple logo. All that you should do is connect your device to your computer, open Redsn0w, choose Extra after which click Just boot. Follow instructions and your device will boot up jailbroken again.