...お出かけ用のワンピースを着て、コートをはおりながら...
...she put on a dress suitable for the trip and while putting on her coat...
I've not seen the verb はおる before and it does not appear in any of my more basic dictionaries. Is it a commonly used word for putting on an item of clothing or is it a more obscure word that was used to avoid repeating 着る? Basically, would I sound weird if I used it in everyday conversation?
For what items of clothing can it be used?
はおる and 着る are somewhat close but not the same words!
はおる(羽織る) means to wear clothes over another, as if you're covering them.
So I'd say someone "はおる" his/her coat when they do so without closing its zipper / buttons.
It's kind of hard to explain in words, but there's a very similar word 羽織{はおり}, which is a noun. 羽織 is similar to coat. Try googling how they look like.
Notice there's no zippers or buttons or obis closing the front? The word はおる originates from this kind of clothing. It is to wear clothes like 羽織. So you can't use はおる against clothes whose 'front' can't be kept opened (T-shirts for example).
And as long as you're catching the meaning of はおる which I explained, it is fine to use はおる in everyday conversation.
はおる is not rare at all, and it's certainly worth learning. (You may not have to remember how to write its kanji, 羽織る)
はおる is more "specific" than 着る. It can be only used for something that loosely covers you from your back. Coats and jackets are something you can はおる, but T-shirts, one-pieces and turtlenecks are not something you can はおる.
はおる(羽織る) is a pretty common word.
It's used for wearing something that hangs somewhat lightly or loosely over your body, on top of inner layers of clothing.
Can be used with: コート(coats)、ジャケット(jackets)、カーディガン(cardigans)、ポンチョ(ponchos)、シャツ(shirts), etc.
Cannot be used with: 下着(underwear)、ズボン(pants/trousers)、Ｔシャツ(T-shirts)、etc.
