I've been told by Inyo N.F. staff that with the switch to a new reservation system last October, now this year people like you should be able to reserve both entry and exit permits on the 6-months-before entry date.
And that may be why you are seeing all the TC exit permits totally reserved already -- by people like you getting their entry permit, and also reserving the TC exit farther out than 6 months.
You should give it a try before Feb 25 and see how far you can get with the Entry + Exit pair of permits. You can always cancel the reservation before paying.
Good luck! ...and please report back how it works out.
I just went on to recreation.gov and I can reserve a permit to enter on 8/11/19 (6 months out) and then I can choose an Whitney exit as far as out as 8/17/19, but that won't be enough time. I would need 10 days between enter and exit.
One strategy is to just snag the South Lake JM21 entry on the correct entry day, choose Cottonwood as the exit, and then when I pick up the permit in Bishop get there way early and try to change my exit to Whitney. But if I don't get it, there's no way I can carry enough food to get to Cottonwood so that sounds like a really risky thing to try.