Previous posters made some excellent recommendations for you...
A question OP...What brand and caliber of rifle do you hunt with? Because your boy might like a "youth version" of it. If the brand doesn't make a "youth version." Buy the rifle... Take off the "original stock" and save it, buy a 2nd stock, cut it down and add a soft recoil pad to fit the rifle to him. So he can have his own rifle. just like Dad does...
except in .223, .243, 6mm Perhaps. When he gets older/taller you can put the "original" stock back on to have an adult size rifle.
Showing him how to hunt the right way, how to clean and preserve his kill and eventually it's what y'all will be eating at the table. So he understands. No matter what his situation is when he grows up. You'll have taught him a valuable skill and with that skill : No matter what happens in life He'll always be able to put some food on the table so the family can eat. And a fond memory of you 2 going hunting/shooting together.
Another option...later on of course.
After his successful completion of a Louisiana Hunter Safety Course. You can "reward him" by taking him to the local gun shop and let him pick out his own Rifle.
Best of wishes for you OP and your Son.