IPhone Redsn0w Jailbreak for iOS 6.1.3 won't work?

I have jailbroken quite a few devices before, so I know the basics of jailbreaking. However, whenever I try to jailbreak my Verizon iPhone 4 with iOS 6.1.3 installed, it works but doesn't work.

I have downloaded iOS 6.0 (filename iPhone3, 3_6.0_10A403_Restore) and done everything correctly within Redsn0w. However, after the jailbreak process finishes (which it does, seemingly without problem), and after I do a tethered reboot, Cydia does not appear.

I'm using a Mac, if that eliminates any possible problems.

I have tried and retried this about 5 times now, obviously without success.

Any help would be appreciated!

Well first of all… A 6.1.3 untethered jailbreak hasn't been made and will most likely not be made.
Apple changed something in 6.1.3 that makes it much harder for people to create jailbreaks. Many of them have advised people to not update to 6.1.3 or 6.1.4.iOS 7 is the next jailbreak they're going to do.