Notification of new employer to current employer (non-compete)

Location: New Jersey

I signed a non-compete agreement with a company based out of California (I am in NJ). I am leaving that company and they are asking me to disclose who I am going to. I would prefer not to but it seems like the contract spells it out plainly:

Should I obtain other employment during my employment with the Company or within twelve (12) months immediately following the termination of my relationship with the Company, I agree to provide written notification to the Company as to the name and address of my new employer, the position that I expect to hold, and a general description of my duties and responsibilities, at least three (3) business days prior to starting such employment.

Do I have any recourse here? I am moving from a company that provides services to clients to a potential client (though we have never worked with that client before). I am concerned with my current employer reaching out to my new employer (I completely forgot about this agreement and didn't mention it while interviewing that I had a non-compete).