Check all neck screws are tight to the body and that neck is stable and won't shift. Check the nut, and this is most often the culprit. Many loose or badly adjusted things along the line can cause tuning instability. Loose string tee screws, loose bridge saddle screws, Loose tremolo claw screws, loose tremolo bridge mount screws. Possibly even lose tuner screws tuners.
Take your neck to a pro, if you can find one. let him do a thorough set up. Neck holes , and even body neck mount holes may need to be readjusted, filled and redrilled. Neck may need to be shimmed.
Good Guitar techs are like good mechanics- HARD to find, easy to get ripped off.