Common:
* Coat of arms: A heraldic design representing a family or organization.
* Coat rack: A stand or piece of furniture used to hang coats.
Less Common:
* Coat-tail: The trailing part of a coat.
* Coat-hanger: A device used for hanging coats.
* Coat-of-mail: A type of armor made of interlocking metal rings.
Figurative/Idiomatic:
* Riding someone's coattails: To benefit from someone else's success.
* Coat-and-tie: A formal dress code.
There are also some less common, more technical words that start with "coat" like "coat protein" (a protein on the surface of a virus) or "coat of varnish" (a layer of varnish applied to a surface).
Let me know if you'd like more examples!