Mieten is completely fine, and you will usually use it without a reflexive pronoun (mir), because the usual sense of mieten is to rent something for your own use. You can insert the reflexive pronoun if you want to emphasize that you did rent the car for your own use, to satisfy your needs. The same is true for leihen — “ich habe das Buch ausgeliehen” just describes the act of borrowing (as opposed to buying/renting/stealing it), or the fact that you're not the book's owner (“ich habe es nur geliehen”). “Ich habe es mir ausgeliehen” means “I needed it, so I just took it, planning to return it after usage/as soon as possible”.
You can insert other verbs, too, depending on the focus of the sentence. In your case, mieten is the most neutral and correct choice. Other choices are:
- leihen would imply that there was no obligation to pay for the car's usage (but often people don't distinguish between these two alternatives);
- nehmen leaves open the legal details (payment) when used with an indefinite article (“einen Wagen nehmen”). Otherwise, with a definite article (“den Wagen nehmen”), it would usually imply that you used your own car: “Bist Du mit der Bahn gekommen?” — “Nein, ich habe den Wagen genommen”, but not vice versa: “Ich habe den Zug genommen” doesn't imply it was your train — you would use the definite article here anyway (selecting the definite vs. indefinite article is a problem on its own).