Amie,
Some permits will show up as available throughout April. Any permits that winners decline online will show up on Recreation.gov "within 24 hours at random times" as it says on the Inyo N.F. Whitney Permit Reservations page. From that page "Timeline" section:
Quote:
April 1 - Reservations open for remaining space and lottery acceptance begins.

> Onsale opens at 7 am Pacific Time (No phone reservations)

> Acceptance period opens for 30 days. Cancellations and declined space will be added to the reservation calendar within 24 hours at random times.

April 30 – Last day to accept and pay for lottery reservations (closes at midnight Eastern Time)

May 1 -All unclaimed dates are canceled and released for web reservations (within 24 hours at random times).


The April 1 7 AM PDT scramble was the ONLY time that the fastest fingers got the permits. From now on, it is whoever is most persistent in checking for permits that will pick up any online permits. Even on May 1, those permits not secured will be released throughout the day, NOT at any given time.

So, you need to create a bookmark/favorite in your browser that you can check often. Make it your "home page" even. Here is a link for June 15 for you to copy and use. You can alter the obvious date so the page opens exactly where you want it. (Note that rec.gov seems to ignore the group size parameter, so it will prompt you for that, but the link will certainly make it easier to find what you want.)
Here is the link to copy:

http://www.recreation.gov/permits//r/entranceDetails.do?arvdate=06/15/2015&contractCode=NRSO&parkId=72201&entranceId=315790&pGroupSize=1

Same as above, but clickable.