Finally, a question that I know something about!
My answer varies depending upon the disaster in question.
In your mentioned case, a tornado, I would suggest thay you go heavy on the first aid, notably bandages. You are likely to encounter a number of casualties, including possibly yourself. You will be no help to anyone if you are injured too. Throw in a poncho, flashlight with good batteries (change annually), a Leatherman multi-tool (accept no substitutes!), some hi energy food bars, hard candy (for instant energy), have a good canteen at the ready (I use a GI canteen with the SS cup and keep it on a GI duty belt), a Lifestraw ($20 but totally worth it), your cell phone and a spare charger. I also keep a pack of Marlboros and a lighter, even though I do not smoke. Others probably will benefit from a smoke, however.
I also keep a BOB handy, with changes of clothes, a sleeping bag and mat, food and water (in a Camelbak), a good fixed blade knife, a small carbine and sidearm in .40 S&W, plus holsters and spare magazines, a portable ham radio (yes, I have a license) that also uses public bands and NOAA weather, chemlighs (changed every other year for new), another poncho (which can also be used as a pup tent), plenty of paracord in OD green, a personal trauma kit which includes suturing kit, antibiotic ointment, bandages, a few band aids, Bactine, iodine wipes, alcohol wipes, nitrile gloves, a biomask, a Mk II gas mask with two spare (new) cartridges, several types of minor surgical tools, all stainless and sterile and more that doesn't pop readily to mind.
I keep a medical turnout bag next to the BOB, which contains a TON of stuff. I bought a full kit and then modified the contents to suit me.
This is the originl kit. I added some other stuff to suit my level of medical training which is similar to yours. There is a higher quality stethoscope, blood pressure cuff, extra meds (they only give ya a few aspirins!!!!!), more bandages, tourniquets, Israeli bandages, activated charcoal and more, plus there is a more useful field surgical kit which includes sterile saline, infusers, tubing kits, scalpels and extra blades, hemostats, clamps, suture kits and a handful of syringes plus a BIG one to deal with pneumothorax. There is a lot more but a list would bore most people.
I plan to be able to bug out in an emergency in less than five minutes with my bags, weapons, computer backups and my portable safe (has important papers in it).
Now, who wants to be on MY side during the zombie apocalypse???