Shave some curls from a piece of wood that has a similar color to the coat rack using a plane. Dampen the curls, and lay them beneath weights to flatten. Remove the curls when they have completely dried.
Cover one of the shavings in clear wood glue and push it into the crack. Place a shaving without glue in after. Alternate between shavings with and without glue until you have filled the crack in the wood. Drip glue into the gaps between the wood curls, and allow to dry.
Chip off any rough edges created by the wood shavings using a chisel. Sandpaper the wood until it is smooth and even.
Add wood filler with a plastic applicator to the crack and the area around it. When the filler has cured, sandpaper the surface of the coat hanger and apply wood stainer and finisher.