Instead, use a large jar or a Nalgene, or even a plastic pitcher. You want something with a top that is relatively narrow and ideally something with a lid.
They have actually invented portable bags with sturdy rings that people swear by (I would definitely get some for my car if I had kids or if I were a Lyft/Uber driver).
The general idea is that it's so much better if you can keep your nose out of it and also not worry about splashing.
It's also really convenient to be able to put a lid on it, especially if you're in the car or in a waiting room. (the bags have a way of sort of closing up, which I don't quite understand yet)