My Cat

So, I don't know what to do with my cat. He is constantly getting out to roam and bringing "gifts" to the doorstep. I put him in a room when I need to go outside for anything and he escapes and like lightning is out the door and gone. He has been neutered and I'm taking him to the vet next month to see if they didn't do it right or if there's something wrong with him. I've had cats all my life, some only indoor, some indoor and outdoor and I've never seen anything like it. I know its instinct to run and climb and hunt, but neutering should have calmed most of that down and he is worse now. He's well fed, he gets played with all the time and nothing burns out his energy unless he plays outside all day, likely murdering multiple little critters. I've thought of giving him to my aunt, she has a farm and he could chase mice all day, but he's spoiled, he doesn't eat his kills, he eats his cat food and he likes to come inside and sleep wherever he feels like and get his cuddles and love so I don't think he would do well as just a barn cat. I'll talk to his vet at his appointment and see what my options are. I'm just worried. I don't want him to get hit by a car or attacked by another animal because he's too big for his britches, or taken to animal control because he is constantly losing his collars (we have spent over 60 dollars on new collars and ID tags for this cat), and I don't want him bringing some disease in the house from rodents. I'm stuck.