You never know the exact time it will drop. The registry starts releasing them at random at the time I mentioned, and since there are usually 30k+ domains on any given day the drop can take at least an hour, and if there are a lot of domains dropping that day it can take two or three hours.
The only thing you can do if you want to register it by hand and you don't have software is to keep performing the search over and over again until it shows as available. I wouldn't try it at Godaddy though, because their checkout process is long as they try to upsell you on a lot of different products. I would try it at Dynadot or Moniker.
The big three you already backordered at are extremely good at catching, so I wouldn't bother trying anywhere else. The only other one was ClubDrop, but they sold out to NameJet, and eNom's registrars that were being used to catch for that service are now used to catch for NameJet. You've got your bases covered.
Just so you know, unless you remove the backorders, there's no point to try to catch it by hand... you have zero chance. Just wait for the big boys to catch it, and hope nobody else bids you up.