If you work you are covered by the payments you have made.(In the case of the OP apparently a year) If you are made redundant or are sacked (but not if you resign) you then get Assedic for a period (about 7 months minimum but it can be longer) and the Assedic pays your social security charges. After that you get gradually reducing cover until you start to pay again.
At no point in this process (say 2 and a half years) would you be in the CMU. I am neither a tax expert nor a lawyer, but I have been there and have the T shirt. I am not, never have been and never will be in the CMU. That is often the case with people who work or run a business in France.