THat's a complicated question. You should call the wilderness permit phone number 760-873-2483 and ask the question.

I think that you have a special case situation, since you have the Cottonwood Pass without Trail Crest permit. It may be that you need to log onto recreation.gov when the last "walk-in" exit permits become available, and then change/modify your existing permit by adding the TC Exit. But that is only my imagining how it works.

Good luck!

...if you are successful, please share how it works. Thanks.