Continuation coverage as defined in section 223 of the tax code is continuation coverage required by federal law. (I'm not aware of any such coverage other than under COBRA.) A new policy would not be federally required continuation coverage.
Since funds in an HSA grow tax free as long as the money in the HSA is eventually used for qualified medical expenses, spending the HSA funds on medical expenses now, including COBRA payments, might not be the best long-term strategy anyway.
