There are several days in August with First available.

I travel this route often. I usually purchase the SFO-JFK portion and use miles for First on the JFK-SFO portion. I always find a First seat available a day or two before departure so you might use this strategy.

I don't think that Plat members have any extended award availability although, one time I wanted to get home and didn't see an award seat. I called, and the agent was able to get me out the same day. Don't know why I couldn't see that online, but it worked for me.

Don't hold your breathe for business class to open up. Rarely see it.