No. RedSn0w was not updated for "true" support on any iOS 6 version above 6.0. If you want to jailbreak 6.0.1/6.1/6.1.2/6.1.3 for the pre-A5 devices, you must use the 6.0 IPSW. The main reason why it wasn't updated to truly support 6.0.1/6.1/6.1.2/6.1.3 is because they were all delta fixes, none of them posing a huge change to the original iOS 6.0, so there was no need to go through the hassle of implementing true support for those iOS. Traditionally, you would use an IPSW that matches the iOS that you're looking to jailbreak, i.e. 5.1.1 for 5.1.1, 5.0.1 for 5.0.1, etc etc. iOS 6.0.1 and above is the exception with RedSn0w.
However, if you would like to use the 6.1.3 IPSW, you can use Sn0wBreeze to create a custom jailbreak IPSW to restore with in iTunes.