In the US, everyone has a common law right to change their name "by usage and habit". That is, you don't have to pay any fees, obtain a lawyer, or get a court order. You just start calling yourself "Jane Doe", and if you do it consistantly and habitually, your name is now (legally) Jane Doe.
To get a Social Security card with the new name, you have to bring the Soc. Sec. Admin. evidence of your old name (eg. your old driver's license) and your new name (eg. a health insurance card).
However, your question specifically deals with the traceablility. I would think getting a new Soc. Sec. card would leave a trail someone could trace, if they looked hard enough. But they would have to know what they were looking for, and how to get the information. I don't think you can just write the Soc. Sec. office and ask, "Did so-and-so change their name, and if so, what to?" Changing your name this way would be much less traceable than going the court order route. Hope that helps.
- Vicki