TheChad23:
OK Lets start from the beginning. Once the dog is down you must freeze it asap. The longer the dog stays out of the freezer the more opportunity bacteria has to damage the skin. The only reason hair falls our of a skin is due to excessice bacteria growth on the hide.
The proper way to preserve the skin starts with thawing the hide and when thawed you will flesh the meat and fat away from the skin. If not all the fat is removed bacteria can build up and damage the hide.
Next Salting.
Use allot of salt and thoroughly rub it into the hide. This is probably the most misunderstood step in the tanning process. Did you salt your hide?

? How long did you allow it to dry?

?
Once salted you will notice allot of moisture seeping from the hide.. This is normal and the hide should not be allowed to "soak" in this moisture. Use a piece of plywood and prop one end up allowing this juice to run off the hide.
Normally two salting is adequate but three may be necessary changed after the salt gets wet looking.
Next dry the hide.
I have a rack and I drape the hide skin side out" over the rack and allow it to thoroughly dry until it's white and hard. This will "SET" the hair and your all ready to tan. I allow most of my hides two weeks to achieve the white hard stage. Most taxidermist's will ship their hides in various stages of white hard but since I do my own tanning on deer and elk I allow them to thoroughly set before introduction to a pickle.
Pickle:
I'll bet you didn't all a bactericide to your re-hydration bath. which is necessary before introducing your hide to the pickle. I use a Lysol or other commercial bactericide to kill off any bacteria before I pickle a hide that way the dormant bacteria doesn't play hell with the PH of the pickle. a re-hydration bath consists of water and Lysol and one handfull of salt per gallon of water. Remember this stage should only last one hour. If your hide takes longer to re-hydrate than use a relaxer like enzol b to help soften the hide. The reason this should go no longer than 1 hour is because it is harder to control the PH in this environment.... Rinse and than pickle.
Keep the pickle PH below 2,5 with the skin is in the pickle. Adjust the water, salt and acid accordingly and test every 12 hours until stable. NEVER allow the PH to go over 2.5 or your asking for trouble.
Degreasing
This is mandatory with any carnivore like a coyote. If you didn't add a degreaser to the pickle and even the tan that's why the hide came out greasy.
Neutralize:
after shaving neutralize in baking soda and water until the pH is 6.0 or higher. That way the skin will accept the tanning agent.
Tanning. I'm not sure what method you used so all I'll say is follow the manufactures recommendation to the letter and all should go right.
Let's review
Freeze
Thaw
Flesh
salt
dry
re-hydrate
pickle... keep pH below 2.5
shave
neutralize
tan
wash rinse and tumble dry to break
If you did all this you should be in good shape. If you skiped even one step minus the washing you probably caused excessive bacteria growth and that's why the hair slipped.
Good Luck
Kuduman