Posted on another forum:

Forgive me, as I'm still pretty new to this process. My permit application was rejected last month, and now I see some attractive week day dates that are still available. They are accepting new applications beginning the 28th (Wed) for those left over permits. Is the process the same, as in, do you send in the same form? Is it still ok to add alternate dates in case your first choice is full? Thanks in advance.

The dates available page published by Inyo National Forest is here:  Mt. Whitney Trail Reservation Availability  Note that the page has not been updated since April 1, and May, June and July slots have been available for call-in since last week. August and later can be obtained starting Wednesday, 4/28. Be sure to call early!

Unused slots can be obtained by phoning the Wilderness Permit office at 760-873-2483, M-F, 8-4:30.

Alternately, you could fax your request to the office at 760-873-2484, but then you have to wait to find out whether you got the dates you asked for. The lottery permit form would probably be ok: It can be found here.  The general application forms can be found on this page.