This is just a suggestion for folks considering applying for a KH scholarship.
IF you think you are a very strong candidate for the KH, I would consider waiting until near the KH deadline in September to submit the KH application and your SMS secondary application. Your secondary essays will get stronger over the next 6-7 weeks as you work on them and complete them. And so waiting is best. It will also provide time to refine your overall package for KH/Stanford. To my mind, and from my limited experience, there is no disadvantage in waiting.
IF you are a marginal KH candidate, I suggest bagging the KH application and submitting the Stanford secondary very soon. Because S has rolling admissions, I believe the likelihood of an II increases the earlier you apply.
Again strong KH applicants are not disadvantaged by waiting until the KH deadline, imo.