Full Time Offer at Consulting Client

I am a consultant of company A (based in CA) working for a client. I have worked for the client in IN through another consulting company B (based in CA). Client doesn't know about A. After that project was completed, the client asked me if I was interested in working at another location in Niagara Falls,NY and now I am working in NY though yet another company C (based in NY). Now the client has offered me a full time position in CT which got opened recently. I have a non-compete with company A that states as:
---
Section 6:
The employee expressly covenants and agrees that during the term of his employment whether under this Agreement or elsewise and for a period of 2 years thereafter, he will not, directly or indirectly, or through a competetior of the company, provide or attempt to provide, any services to customer of the company without the specific written consent of the company.

Section 19: Termination

This agreement may be terminated by the employee if the employee gives the company atleast 30 days written notice prior to the termination date.
---
Company A that I recieve the paycheck from is in CA and the client is not aware that I work for A. As far as they know I work for C (B before that). Can I take up the position? Will the 30 day notice termination tetminate the Non-Compete as well?

Also the client said that they contacted C informing their intentions to take me full time and C said that the client cannot hire me. I have never signed any paper with C. Does C have a right to say that?