There are uniforms today that have a "wicking" action that can actually "wick" perspiration away from your body.
However, on very humid days/evenings on the ball field when the surrounding air is very humid, there's really nothing that these uniforms can do. Even products that assist in the cooling process can find it hard to manage.
An alternate is to use two things - clean T-shirts dusted inside with talcum powder (baby powder), and a dusting of talcum powder on a towel.
The T-shirt change after every inning of work can be very refreshing and simple to do. You just have to remember to pack them in your bag.
The towel dusted with talcum powder is a refreshing "after shower" application that can bring you around after a hot and humid day of work.
Below is a picture of how to apply and use the combination of talcum powder and a towel. The human back has a large area of skin and sweat glands. Thus, dusting your back is not only refreshing, but long lasting too.
The picture on the right is the method to use with the towel. Although the example shows a person with the shirt on, right after your shower when you're all dried off, is when you apply the talcum powder on the towel. Obviously, directly after showering you won't have a shirt on.