Hey fibber,
First up, it does matter which round you apply in. The earlier you apply, the better your chances are. In later rounds, the competition gets tougher as the seats are lesser.
Having said that, there is no point in rushing your application to an earlier round, just to make the deadline. In an ideal world, you should have applied for round 1 with plenty of time to spare. But it's neither an ideal world, nor do schools get all their candidates in round 1. Schools limit the number of people they pick in earlier rounds based on the quality of candidates they get. The other rounds exist because schools haven't been able to fill up their seats with quality applicants. The admission teams have a said number of rounds, because they predict that they will be able to complete their quota only by the end of all the rounds.
So in my opinion, its better you push your application to a later round, if you feel that you can significantly improve on your application with the extra time you have.
The reason why Round 1 applicants find it easier is because, a lot of applicants who plan on applying in round 1 give very little time to the different components of the application and later end up realizing that certain components take longer than expected.
You can check out the different school pages in our website to understand what the application process is like and how much time it would take you to complete the applications. We also provide school deadlines for different rounds primarily to keep candidates informed and make all the admission information easily digestible. You can check out our website in the following link:
www.mim-essay.com
Hope this information was helpful. Cheers.