Really, you mean while driving there? How many unsterilized hands have held that steering wheel before you?
I don't put mine on until I'm going to do something where I expect to encounter bodily fluids. Meaning I (GASP!) touch the patient without gloves.
To the OP, scene safety starts at dispatch on a particular call and doesn't end until the patient is delivered. Honestly though, you should have a safety focus from the time you clock in till you leave the parking lot.