If they continue to adhere to prior practice, you may be able to get your extra day permits in June a few days early, IF they actually have a glut of permits. Normally in June they would have a glut; however, with the low snow, this year could be different. Alternatively, if you monitor recreation.gov, you may be able to pick up the extra permits ahead of time that way.
I have to think that the ONLY way to do two back-to-back day trips into the Whitney zone is by having two back-to-back day passes. I don't think the kind of trip you have in mind is the type of abuse that they are concerned about (i.e, as I read your question, you are not trying to use a day pass to get a jump start on a multi-day/night trip for which you have another permit). You could call them to be sure, of course. In the past, I have gotten accurate information over the phone at (760) 876-6222.
