It turns out that the 1.1.3 jailbreak instructions from iPhone Atlas is pretty much the best 1.1.3 instructions I have seen around.
I’m finishing the final step of the jailbreak right now, after a bit of difficulty. It turns out the difficulty (as well as the problem I experienced earlier with Nate True’s method) wasn’t something I was doing wrong. It’s because *.nyud.net (Coral Cache) doesn’t work on the office network, and the iPhone haxorz, in an effort to save bandwidth, are hosting a few of the packages over there. The soft-update package from the other day didn’t show up because of this. The OpenSSH package today didn’t show up because of this, either. I ended up having to install the VT-100 terminal, then use a little
sed magic and regular expressions to remove Coral Cache from the URLs. Things as slower, but things are now actually possible.
EDIT: It mostly worked, except the cellular service was dead. I had to follow these instructions then resync with iTunes to turn it back on again: http://guides.macrumors.com/Jailbreaking_and_updating_to_1.1.3#Post-jailbreaking_-_Restoring_your_phone_service_in_1.1.3