there is prop called keyboardVerticalOffset that you can pass to the KeyboardAvoidingView that will change how much the keyboard moves past the textInput.
Sample of my code:
const keyboardVerticalOffset = Platform.OS === 'ios' ? 40 : 0
return (
<KeyboardAvoidingView behavior="position" keyboardVerticalOffset={keyboardVerticalOffset}>
<ListView .../>
<KeyboardAvoidingView/>
)