Show the whole rack and was this new or used.

Am not sure what is going on in the photo. My suggestion is to screw the coat rack directly to the wall. Use a stud finder to locate studs and anchor the rack directly to them using wood screws. This should provide maximum stability. Good luck!

Use a pan head screw and the screw will enter through the half moon slot and the horizontal piece will then catch the head of the screw. Don’t use a bugle head screw like a drywall screw as it will have a tendency to pop off. If you are uncomfortable with this get two D rings and screw them into the back of the board and use them to hook onto the screws in the wall

Thank you for your suggestions. I realize that I should have included more specifics in my original post. I’ve a couple more pics for reference. This is a used coat rack I purchased. In the new pics, you can see I already put anchors in the wall. I experimented with adding washers (about 6 of them) to the wall screw for a snug fit into the holes on the back of the coat rack but it didn’t work because of the angle needed to get them to fit into and behind the bracket above the hole. Hopefully you can visualize the problem because it’s hard to explain. The holes measure 1 1/4” diameter and about 1/2” deep. I’m hesitant to screw the rack directly to the wall because I would have to drill through the front of the rack. The pan head screws seem like a good option but how would I prevent the rack from moving side to side since the hole is much wider than the head of a screw? I appreciate all your feedback!
Thanks, Kim

You might try looking for flush mount clips...interlocking hangers reminiscent of french cleat fasteners. Remove the flat plates on ends of the board and replace with the new clips using flat head screws.

Screw corresponding clips to the wall. Lee Valley has these clips on page 227 of their

Hardware catalogue...very inexpensive and very versatile;sturdy and easy to dismount.

I'm not sure what you're asking, but they hung the rack by two nails or screws with heads. The metal plates would set on top of them, allowing the rack to be nearly flush with the wall. Measure the distance between the centers of those holes -- probably 16", a standard wall stud separation.

eta, not sure how they kept it from sliding. You could notch the metal plates... or better yet, put a dab of hot glue on either side of a 'slot' for your nail/screw shaft.