My cat was like that in the last couple of years of her life. I took her to the vets, they did tests and it turned out she had thyroid issues (I forget if it was overactive or underactive). The vet prescribed her medication and needed to see her every few months to take blood to determine if the dosage needed to change.My cat hounding me for food, no matter how much I feed her she will eat it and still hound me. I think she has hyperphagia. I have a huge ceramic dog bowl that is filled to the brim with dry food and it's never empty so there is always food here, she also hunts. but no matter the time I get up or even if I so much as looked like I am going to go past her wet food bowl, there she is hounding me. even if I have just fed her half an hour ago.
The medication didn't fix the issue, it was just to control it. She still wanted more food than she needed, but it did improve things.
Yours might have a similar issue?