Yes, but the COBRA enrollment period has likely elapsed. If it hasn't, this is definitely the best option. In addition, if the OP was entitled to medicaid or any other services due to being uninsured, he has probably missed those, too.
Now, if the patient is expected to pay back the discounted rates paid by insurance less the cost of his premiums, there will be a measure of savings there, but I doubt that is what will happen. The insurer will probably just take back the entire amount from the providers/hospital/labs, etc.
Depending on OP's initial condition, there was probably less of a hassle getting care initially because the hospital thought he was insured, so there is value in that. Otherwise, I do think there is some responsibility on the employer's part to try to fix this mess.