There are some decisions you will have to make.

Are you going to leave the island or stay?
If leaving, would a bicycle, cart, or other mode make it easier, more efficient?
Where are you going to? A predetermined destination is very important.
How long do you think it is going to take to reach your destination given gridlock, bad weather, and masses of other people fleeing?
If the family is split up, where are we going to meet up?
Where are the secondary meeting points?

You have reached the point that all people who prep eventually reach. My pack is too small to carry everything I want/need and I don't have a better destination to run to. What am I to do?

You either decide not to leave and make your primary residence better equipped and you get an alternate destination and begin equipping your self to travel as well as increase stored supplies at the destination.

Most people who frequent this forum have moved from 1 giant backpack with everything possible to many smaller equpiment containers in a layered approach.

They have a few items that they carry on them all the time. Knife, firestarter, whistle, money. Then they often build a small kit, the Altoids tin seems to have many fans, that has a few additional items to help you GATHER other supplies to allow you to SURVIVE (not necessarily thrive). Often the next kit is a day pack. Large enough to have some shelter, food, water, light, heat, cooking. It is portable, small enough to carry even out of shape. It is not seen as Rambo-esque by others since so many people use "book bags" and day packs. You can easily have several, one for each family member. The next layer is usually a large backpack or tubs. Tubs can be moved with vehicles to the destination location.
Each layer usually has similar items as the smaller layer for redundancy. Additionally there can be office and vehicle kits. The idea is that you are always near at least one level of kit with the hope to get to the next kit and home.

What should you have in your kit? Which kit are we talking about? It depends on what kind of problems you foresee for your area of operations. Power outage is different from Twin Towers disaster. Power outage you need alternate heat, light, cooking and can stay in your apt. Twin Towers and you need a dust mask and running shoes.

The basic items for survival are Shelter, Water, Food, First Aid, Knife, Fire, Signaling. A space blanket will provide shelter for $1. A plastic trash bag can be worn as a poncho and provide shelter. My fanny pack kit has a space blanket, my day pack has a tarp. My home kit has multiple tents and tarps.

I would suggest you pick the container you feel comfortable carrying first. That will help you plan your purchases and supply choices. I felt a fanny pack would be a size I would carry. The size dictated would I could pack. I could not carry gallons of water but I could carry purification tablets and baby bottle liners. I could not carry a tarp but could fit a space blanket and trash bag. LED lights with watch batteries fit, 2D cell flashlights don't.

If you feel you will be walking miles home, then you need to have a pair of comfortable walking shoes stored at your work. If you will be riding the subway, then you need to have alternate lighting. Cash can get you needed supplies assuming you act quickly before supplies are depleted.