Am I able to get a new CAC after my ETS date?

When I was in a similar situation, I was able to take my orders to my new unit to get my new CAC. If you have such orders, then it should be possible. It would also be possible without orders if your new unit already have you in the DEERS system under their UIC or however it is arranged in the RAPIDS system. Check with your gaining unit UA (unit administrator) or S1 if you don't have new orders yet.

Edit: Are you leaving active duty and joined the Reserves? That is how I read it. If you ETS'ed from the Reserves, is there another reason you would be entitled to a CAC?

No, that is why it had an expiration date.

It depends on what your DEERS status is.