Maybe I'm getting cynical with age, but ....

Part of being a mechanic or, Auto Tech", as they are called now, is dealing with customers. Right, wrong, or indifferent, the customer has more control over your business than you can ever imagine. A customer who receives great work from an auto shop will usually tell two or three people IF asked. A customer who receives poor work, or what they consider to be poor customer service will tell everyone they meet that day WITHOUT being asked, and will bring it up again in any conversation that comes up regarding auto service. and they will do that whether you (the Shop) is right, or not.

Dealing with a customer who thinks you are responsible for something, even if you are not, is a SKILL. Having that customer leave with a smile on his/her face after such an encounter is an ART. It's tough, and you usually end up swallowing a lot of pride.
You don't have to lose money, or business on the deal, though, and that's the important part. You do have to invest some time, though, and that's always a difficult idea to accept, especially if your very busy.

It is a necessary thing to do, though. Taking time to calmly explain the the work you've done, and how it doesn't involve anything that is part of this new issue, in a manner in which they can understand, is key.
It IS possible to explain things logically and calmly to the customer's satisfaction, even if the customer insists that everything is your fault. It might take more time than you are comfortable with, but USUALLY in the end, the customer will leave satisfied, and you won't have to eat any repair work or towing charges, and more importantly, you won't screw up your company's good name or reputation.

Complaining customers what one thing, attention. They want to know that you are going to listen to their complaint, and what they think the cause and the remedy is. Usually, after they've gotten it off their chests, they are more likely to listen to reason, but not excuses. Be honest, logical, calm. Explain things and stop if they have questions or comments. Don't try to talk down to them, and don't try to throw a blanket over the situation and either ignore them or tell them flatly that they are wrong. Even when they are.

Usually when it comes to care repairs, people are spending a lot of money, that is worth a few uninterrupted minutes of your time. You'll find that you're interest in them, and your patience, will pay off in the long run.