Yes, you can schedule your eligibility period as soon as the same month of application.
For example if you are in April you can apply for April-May-June eligibility and certainly you can schedule for May-June-July.
The restriction is not about how soon your eligibility period but it's about how late it is. They don't let you schedule more than six months before the intended eligibility period. So for example in April you cannot schedule for October-November-December.