Get it a collar with a bell, and if the cat is exceptionally prolific, add another bell. A collar with a quick-release buckle, or better still, those which are all nylon - like a stretchy piece of elastic, so the cat won't get caught up in branches. RSPB says collars with bells cut 50% of bird kills. Mice and other creatures will also have a better chance of hearing the cat's approach.
Have a go at de-training the cat's predatory instinct using the 'aversion therapy' of a water pistol, fired at the cat when it is seen attempting to creep up towards a bird. Cats don't like water, however harmless it is, and when this is linked to a firm "No!" you may be able to uncheck the pattern of behaviour to some degree. I did with a little black cat, one who never ventured beyond his own garden so was usually a 'sitting target' for my marksmanship with the water pistol.