New baby on the way and switching jobs just before baby is born

My wife (30F) and I (30M) are expecting our first child in 2 months (mid-October). We live in Minnesota. We currently have a pretty good family health insurance plan through my work and have nearly hit the deductible with the pregnancy and other things. The current plan is Ind Ded = $500, Fam Ded = $1000, Ind OOP $2500, Fam OOP = $5000. In one month (mid-September), I will be switching jobs (it’s a job type I studied at school for, get to work with some acquaintances of mine, has better pay, better PTO, paid paternity leave right off the bat, way more flexibility, helps pay for childcare, etc.) and my wife is leaving her job as well when baby is born to stay home with the baby for a few months and is then starting her own business (mental health counseling). With this situation at hand, what is the best thing we should do about health insurance? I don’t want to start over for paying a new deductible this late into the year. The one thing my current job has over my future job is better health insurance (much lower deductible). My current job’s health insurance also lasts through the end of the month I resign, so through the end of September. Both current and future job have some flavor of Blue Cross Blue Shield. Right now I’m thinking of delaying starting my new health insurance until the start of November and using COBRA for October, although I don’t know when the hospital sends the final bill for the birth, is it usually in the same month? Or is there usually some postpartum care in the first few months of baby’s life that’s usually wrapped into that bill? Any suggestions/answers you all have?