I concur with @NCperson and would push it to July 1st to be safe.
Keep in mind that if you establish your "new" residence before the closing date of the old one, I don't know what consequences that will have for causing issues should you be audited on this matter. Typically, you move out of the old residence a few days before you actually sell it, at the least. But most home sales I'm aware of, the property was vacated by the owner a month or more before the sale so that the property is clean and empty for showings.
