It depends on the terms in the group policy. Contact your company's HR department and address these questions to them. In addition, you can contact the insurer directly.
Usually, group health insurance plans have a designated open enrollment period prior to the start of the coverage for a year. There are exceptions called Qualifying Life Events as defined in the policy. I do not believe that canceling a month to month policy would be a qualifying life event. Usually it is something like having a baby or those spelled out in the group policy.
Consult with an attorney and follow his/her advice.
You must consult with your HR department to see what your options might be. If the plan has one open enrollment period, as most do, you must wait for the next one and continue to maintain your own policy until then.
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