We had a coat hook screwed onto the solid part of a bedroom door and it served well for about 10 years, but it has now fallen off and the screws won't re-secure. I have considered filling the holes and trying again, but have concerns whether screws into filler would hold the weight. I have also considered using rawl plugs (anchors), but don't know if I can get any small enough - those we have seem too long.

Do I have to abandon the existing holes and have the hook placed a little off-centre with new holes?