Here is below given very easy procedure which is currently applicable for jailbreaking 6.1.3 on iPod Touch 4g. The iOS 6.1.3 jailbreaking is not a rocket science nor it requires any expertise about the software. A common smartphone user who can run a computer can easily jailbreak 6.1.3 on iPod touch.
The thing to remember at this time is that the available solution to jailbreak ios 6.1.3 is not an untethered jailbreak for ios 6.1.3 but it is semi untethered for pre A4 devices which include iPhone 4, iPhone 3gs and iPod Touch 4g. It means you can also jailbreak 6.1.3 iphone 4 and iPhone 3gs by following the below guide.
The 6.1.3 Jailbreak Software
Although evasi0n is the best option to jailbreak ios 6 ( up to 6.1.2 untethered) at the moment but unfortunately it does not support to jailbreak 6.1.3 and 6.1.4 on any type of idevice. The only possible solution is through limera1n exploit which can be injected using the redsn0w 0.9.15b3 in combination with iOS 6.0 ipsw file.
Step Wise Guide To Jailbreak iOS 6.1.3 Semi Untethered
Step 2: Download the redsn0w 0.9.15b3 from this page for both Windows and Mac.
Step 3: Run the redsn0w program and from main window go to Extras => Select ipsw and locate the iOS 6 firmware file which you must have already downloaded in step 1. This is important because the redsn0w 6.1.3 jailbreak is still based on limera1n exploit which is compatible with ios 6.0 otherwise you will get unexpected jailbreak errors.
Connect your iPod Touch 4g to your computer for continuing the jailbreaking process for 6.1.3.
Step 4: After the file is selected successfully go to main redsn0w window and click Jailbreak when asked select Install Cydia and click Next (Cydia is the main program which you shall use to install jailbreak apps and tweaks once the ipod touch jailbreak is completed).
Step 5: The jailbreak software shall ask you to put your iPod touch in DFU mode (DFU is direct firmware update mode which is must for jailbreaking the pre A4 devices), this requires pressing the both power and home buttons in different combinations as mentioned on the redsn0w screen. If you feel difficulty in putting your ipod touch 4g in DFU mode you can follow this tutorial prior to starting the 6.1.3 jailbreak process.
Once you have completed DFU mode the jailbreak for 6.1.3 on ipod touch will start and will complete in few minutes. Device will reboot and now it is jailbroken but you need to complete one last step other wise Cydia and Safari will not work because this is a semi untethered 6.1.3 jailbreak.
Step 6: Do the step 3 again and from main redsn0w screen select Extras => Boot Tethered and follow screen instructions to put in DFU again and wait for ipod touch to reboot.
Now your device is jailbroken and you can use Cydia for installing the apps, tweaks and themes of your own choice.
Remember this is semi untetherd jailbreak for 6.1.3 and if you reboot your ipod touch without step 6 you will not be able to run Safari & Cydia and both will crash.