You can always register the device with FDA, but I think you are asking about a 510(k) type submission. I did a deep dive on this issue for a client last year, never came up with a firm answer.
I think you would have to submit a 510(k) and claim SE to another 510(k) exempt device. I had colleagues who said FDA would accept this and others who said it would not. Those who said yes pointed to examples that, upon closer inspection, were not what they seemed. Those who said no...can't prove a negative. What little information I could find from FDA was ambiguous, I thought.
So far, I have not come across anyone who wanted to front the cost of a 510(k) to find out. Maybe someone here has tried it.
Last year's client's situation was fairly unique, but there might have been another option for them, which they opted not to pursue. I don't feel comfortable getting into the specifics. However, if you want to tell me more about your device, there is a very small (I think) chance there might be another option for you. But, once again, I know of no one who has tried it, so I can't say for sure.