No, you can update your new scores for your application once you get them. Personally, I would apply as soon as the application is available to submit (early next June) and have the new scores sent ASAP as well.
Ah so I could put that on my application that don't process my app until my July scores come in? Like what if they make their decision before seeing the July scores?
No don't tell them not to process it, just put it through normally. If you're not sure about getting an interview with the scores you have now.. maybe wait to put it through? It's your call. Either way, I'd get your application 95% completed by the end of May if possible just so you can submit ASAP. From what I've heard, the earlier you submit (with rolling admissions schools), the better chance you have.