You do not need to write yourself a check. You can deduct the mileage or percentage of actual expenses on your own return as a employee business expense the same as any other employee.
Alternatively, the S-Corp could reimburse you and deduct the reimbursement. If done at the Government mileage rate for only business miles, it wouldn't be shown on your return. If reimbursed under some other method, you would report the expenses reduced by the reimbursement.
One thing that you can't do is let the S-Corporation claim the mileage, as the S-Corporation doesn't own the car.
