What is the best way to improve your cat's coat?

Charlie my cat with his shinny black coat

It is a waste of time writing copious pages of waffle in answering the question in the title because the answer is straightforward and common sense: ensure your cat is healthy by being observant and taking him to the vet when needed and not putting it off, providing the highest quality food you can afford which means high quality wet food or a properly prepared and stored home made raw diet and combing his coat regularly. That stimulates the oils. I comb my cat, who has a single coat, with a flea comb all over. This is a very fine comb (32 to the inch) and he likes it a lot as it massages his skin. I use it myself to scratch my head ;)

A contented cat too will be healthier through lowered stress. That is also down to us in providing an excellent environment.

Fats - essential fatty acids (EFAs) - are important for a cat's coat. Omega 6 and Omega 3 fatty acids help to moisturise the skin. 20-25% fat on dry matter basis is fine (how to read a label on American cat food!). High quality cat food should contain this.

Cats keep their coat in good condition through grooming. Old cats neglect grooming sometimes. We have to step in and help. Fat cats can't get at some spots. Once again we have to help.