iOS 6.1.3 final has finally arrived with bug fixes and improvements over the last release. The bad news is, Apple has patched the Evasi0n untethered jailbreak for iPhone 5 and newer devices.

If you updated to 6.1.3, there is no way to untether jailbreak your device. If you are using a newer device there isn’t an option to downgrade your firmware. However, A4 powered devices can tether jailbreak iOS 6.1.3 with Redsn0w 0.9.15b3.
Devices that can be tethered jailbreak
- iPhone 4
- iPhone 3GS
- iPod touch 4th generation
Devices that cannot be tethered jailbreak
- iPhone 5
- iPhone 4S
- iPad
- iPod touch 5th generation
Download
Step 1: Download iOS 6.1.3 firmware.
Step 2: Download Redsn0w 0.9.15b3, unzip and run the program

Step 3: Click on the Extras button.
Step 4: Choose Select IPSW and select the firmware you downloaded.
Step 5: Click on Jailbreak and follow the instructions. Choose to install Cydia when prompted.
Step 6: When prompted, enter DFU mode.
Step 7: Go and make yourself a cup of tea while your device is being jailbroken by Limera1n.
We have tested and can verify that this tethered jailbreak works. Always backup your device before attempting to modify it in any way. Happy jailbreaking!
Once you have Jailbreak your device you can go ahead and check out the best Installous alternatives.
If you brick your device, you can read about recovering your device here.