With a Target, hunting, Sports person carry license in Westchester, you're absolutely allowed to carry on your person when driving and when coming from or going to whatever your permit allows you to do. It is a carry permit, and not a premise permit. With a premise permit it is my understanding you are not allowed to carry it on your person. You have to transport it in a locked box from your premise to the range.
In New York state you are not required to inform an officer if you are pulled over for a motor vehicle infraction. There isnt any reason for anyone to know that you're carrying, unless God forbid you had to use it, or you had some sort of a motor vehicle accident where you were rendered unconscious.
In the unfortunate event that either one of those things happens, you probably have bigger problems to worry about, but if you're only worried about justifying why you're carrying then just make sure you can prove it.
I'm curious what you mean by premise permit? Isn't that just basically target or does it say "premise" on license?