Health insurance issues

I'm absolutely lost on what to do, my husband has provided us with health insurance for a while now with his job. Unfortunately last week his job decided they could no longer afford to keep his position open and laid him off. Our insurance runs out the last day of this month. Now normally going without insurance is not a big deal for a little while, but I'm currently 37 weeks pregnant and baby is due early next month. Shockingly he found a new job already which not only pays better but full benefits start immediately after hire. Now the problem is we have no idea when insurance 100% kicks in due to the fact he has to have background check completed before payroll/first day starts. We may not have coverage the first few weeks of November, so if she wants to come within those weeks we have to pay for birth ourselves, of course we don't/can't afford to. Cobra seems so unnecessary to do for just a few weeks but we may have to just do it. His job knew he had a baby due soon but like most big corporations they don't care what your situation is. Any advice is greatly appreciated! I just wanted to focus on getting my baby girl here happy and healthy not stress about health insurance!