Good heavens, NO !!!!
You drive back to the station where your other set of PPE has been stored in
the approved manner, change there and store the first PPE in the aproved manner.
Then respond (with due caution and awareness of staff allergy to the risk of
being asked Questions) to the second tasking.
If, on arrival, the job requires the other set of PPE, you return to the
station, continuing to assuage staff allergies to risk (of questions), retrieve
the original PPE, change, correctly store the intermediate PPE, and return to
the ashes of the second job.
I trust this has assisted your understanding of PPE policy as writ.
cheers