I would assume getting everything verified would help. Also, sending primaries and receiving secondaries and filling them out earlier. If I remember correctly, schools will send you secondaries and allow you to submit, but won't review until all of your materials are in.
You are correct that submitting June vs July makes no real difference if your committee letter is going to take longer than verification. I had some frustrated friends that worked hard to submit early June only to be waiting for weeks past verification for their letters to be sent.
But, verification starts to take more than a month right about now, so submit ASAP if you're expecting your letter to be sent in during mid-August.