Not both, no. On an individual return, (Form 1040), claiming the standard mileage deduction is a provision that is allowed to simplify claims of vehicle expenses. However, Corporations, S-Corps, and Partnerships may only claim actual expenses for vehicles. Thus, your S-corp may claim depreciation, fuel expenses, oil expenses, repairs, insurance, and so forth. But not mileage, when the car is owned in the corporation's name. (Even on an individual return, you may claim either depreciation and actual expenses or the standard mileage rate, but not both).
However, there is a workaround. That workaround is for you to own the vehicle (Not the S-Corp). You can set up an accountable plan for the corporation, and then you submit a log of your mileage. The S-Corp can reimburse you (tax-free to your personal income, and a deduction from corporate income) up to the standard mileage rate. More information can be accessed in SuperUser Opus17's answer to the question in the following link: https://ttlc.intuit.com/questions/4124147
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