In most states the Uniform Trade Secrets Act would prohibit you from doing so, but New York has not adopted it. A client list is usually classified as a trade secret.
These are just some basic suggestions that you want to run by a lawyer (in other words this is not legal advice), but I would suggest you consider doing the following which may help you stay out of trouble:
1) Take out an ad in a local newspaper and say "John Doe, formerly with XXX has now opened his new office at ***.
2) Only solicit clients from whom you were the primary point of contact at the business.
3) Wait until you leave the business before sending any written correspondence with your new contact information.
4) Do NOT remove any client files, or any other documents belonging to the old business when you leave even if a client says they will follow you.
Again, I urge you to contact an attorney because this is an area where you could get sued quite easily. Also, make sure that your employer never made you sign a non-compete or non-solicitation agreement especially if you've worked there a long time... you might not necessarily remember what you signed when you started.
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How and when the employee got the contact numbers and how and when the employee contacted the customers would be important points in considering whethere there was a violation of a duty owed, interference with contract etc.
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You may want to get a legal advise on this. Sit with an attorney before moving further.
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