First off you CAN check in whenever your site is ready so my statement was not incorrect. Your statement, while obviously not incorrect because you did it, would be much more dependant on how full the fort was. If there are no open sites, there will be nowhere for you to go. Although the original question was the " Earliest time " I was trying to answer more about any day, any time of year, or whatever type of rig/rig requirements that someone might have.My understanding is that this is a very busy time at the fort and availability may not be there at any time you arrive.
Due to your circumstances, they may have gone beyond what would normally be acceptable and may not be something you could count on because you made better time than you thought you would. I could be wrong but, I have never tried it, so I can't say for sure. I do know that we have never been allowed to go to a site until it was cleared through Mousekeeping.
In the future I will try to mention in the post that my response is based on what I have found to be true but, other options could be available exceeding my information...