What can I do about musty smell on fleece shipping halter?

I have an old (but in great shape) leather shipping halter with fleece attached. It’s been stored in a bin and n the garage for a few years and now I want to use it for shipping a horse - but there’s a musty smell on the fleece. I don’t see any mold or anything. I’ve so far used vinegar with a washcloth to scrub it gently and just did it again with a tiny bit of dish liquid and vinegar. It’s drying but I can tell the musty smell is still there. Any reason I shouldn’t use it? I feel terrible because it’s fine other than that and was my heart horse’s halter with her name on it on brass plate.