How do I continue cover when my VLS-TS visa is expiring and I am still waiting for ANEF to process my residency renewal?

There should be a latency period of 6 months between the expiration of your TS and the decadence of your PUMA rights, but it is possible the rule has changed or does not apply to your case for some reason.

After three months of being "irregular" (in the sense not having a TS valid for the securité sociale anymore) you can be eligible for AME, but that would also require your income to be below a certain threshold.

If the CPAM terminates your rights at the expiration of your TS you may have grounds for an appeal.

If your rights are terminated, there is no reimbursement for the period you are without coverage and only the delivery of your new residence permit will unlock your eligibility.

I am not a lawyer so takes all this with a grain of salt, let's see if someone more expert than me can contribute.

Sorry, I forgot. You may try to ask ANEF for an APS (autorisation provisoire de séjour), I am not sure if it is possible and/or easy to get on in your situation, but that document would allow you to keep your PUMA rights.