If a supervisor has to drive a work crew from one state to another, should that work site supervisor also be compensated for driving the company vehicle?

Not necessarily. If a supervisor is salaried, additional compensation is not required. If he's hourly, then he's compensated according to the employment agreement.

Is the supervisor salary or hirely,

If hirely they will be paid for the hours they work.

If salary often this is just something they have to do.

Normally they will of course get their meals and overnight costs covered.