Hunting and Fishing May Not be an Option
When I traveled to assist with the cleanup after Hurricane Katrina I expected to have issues with insects. I expected mosquitoes, flies and roaches to be crawling from under piles of debris. What I found shocked me. There were none. In fact, there were no small animals, birds or insects. They had all died in the storm. There were either washed away by the tidal surge or blown away by the winds. If you plan to hunt for food after a disaster you may not have anything to hunt.
Also, you cannot expect that you will be the only person or thing hunting for whatever food source is left. The demand will certainly be greater than the supply and the predators will certainly outnumber the prey.
There are stories today of hunters who have shot a deer and then tracked the deer only to find another hunter or group of hunters claiming the kill as their own. I can see this as a real situation now and a guarantee of deadly trouble in a true survival situation.
Using something like a pellet rifle or crossbow may be an option to quietly hunt for food.
Nothing Left to Hunt
If the disaster lingers or lasts for months or years it is very possible that the available game for hunting will not have enough time to reproduce to the point where it can be maintained. It is very probable that the supply of game will be hunted out. Do not plan on hunting as a long term solution to feeding your family.
Depending on the type of disaster you may not want to eat what fish you catch due to the contaminants in the water.
Water Polution Could be a Serious Problem
Flood waters in the South Texas floods of May 2015 breeched the walls at sewer plants along rivers releasing raw sewage into the rivers and streams in the area. Dead animals and dead people also were in the flood waters. Vehicles dotted the rivers and streams releasing their chemicals into the water. What fish that survived all of this pollution were probably not fit to eat.
The most foul smelling and polluted water I have ever encountered was right after Hurricane Katrina. Nothing could live in that mess. There is no way you could have pulled anything edible out of the stagnant water in the canals.
You will have to wait until the water clears and the fish return before attempting to eat anything from polluted waters. Even then, how are you going to know that it is safe?