Starlings are clever, I must admit that. As much as I hate them, I will admit that I enjoy hunting them. Kind of like backyard turkey hunting. Here's a few tips I've used over the years that help to pile up a body count without spooking them off for good.
1- Get a squirrel-proof birdfeeder (the kind with the drop-down hatch or such). That will prevent them from stealing directly from the feeder (if you set the trip weight low enough), but other birds will still spill enough seed on the ground to attract them. This sets up a killing field on the ground. If you have multiple feeders, so much the better. Laying out feed corn on the ground works for squirrels, but starlings and grackles seem to know something is up.
2- Use some kind of silenced rifle. If a PCP, get an LDC; If you prefer springers, use a suppressed version. My Gamo Silent Cat is dedicated for just this purpose. Either way, stick to sub-sonic ammunition. Pointed pellets are the best in my experience; broadside shots with hollowpoints and domes have a tendency to bounce off their feathers.
3- Shoot from different spots. If you have to go out your back door and sneak around the side of the house, do it. You might get one or two kill shots from the same position, but they soon learn where the lead is coming from and avoid getting in the line of fire from that spot. Switch it up! Using a quiet rifle may help to hide your location, but even then you won't get many shots without moving.
4- Police your bodies immediately. If you can get a quick two-fer volley, that's great. But be sure to get the fallen out and away from the area as soon as possible. I throw them into a ditch to feed the local feral cats, but a trashcan is probably the more civilized option.
Happy Hunting!