New Mexico LLC
You can certainly form a new NM LLC, and make it the sole member of your older LLC. Depending on the state where your older LLC was formed it might or might not be required to file amendment with the state. In any case you would need to amend the Operating Agreement (if you had one) for the first LLC, but its an internal document not filed with state authorities.
As far as third LLC is concerned, I presume you are wondering if creating lets say a Delaware or Wyoming LLC and have it be owned by the NM LLC and in turn use it to own the older LLC would in fact provide a better protection to your personal assets in case your older LLC is sued. I guess it would, in a similar way as an additional body armor would protect a motorist, but would make the journey more complicated and less enjoyable.
In fact, if your purpose of forming the NM LLC is the famous anonymity with no annual cost other than a registered agent, but you want to add an extra layer of protection, you might as well just form a Wyoming or Delaware LLC, as neither state requires listing members on the Articles anyway, and skip the NM LLC part. Keep it as simple as you can, in other words.