I had similar sitution. My OPT was experied in August and H1 started in Oct.
First of all, you can stay in US during July and Sept 30. There is a 90-day grace period so you don't have to leave the country.
Secondly, you should apply your H1 in April. They start to accept applications on April 1. You really should try to turn in your application ASAP. (April 1 if possible).
Hopefully this message helps.