Having high-centered our rental car on a gravel ridge, near the ghost town of Shakespeare, we have some hard-won and expensively purchased advice. First, when we called Genesis Towing, a recorded message told us to leave a message at another number. Then, when we called Big 'Uns Towing, the dispatcher did not know where Shakespeare was, or the highway number (both are on our maps and both are less than 10 miles from the center of Lordsburg.) instead we were told to call the local auto dealer, Smith Ford (575-542-3552.)
Now, Smith Ford's tow truck driver was quick, polite, sympathetic, and courteous. However we recommend that when the dispatcher asks, "How are you going to pay?", you should ask, "How much is it going to cost?" BEFORE you tell how much cash you have on hand. The total ($150, to drive less than 10 miles, lift the car off the gravel ridge, set it down on the road, and escort us less than 10 miles back to the dealership to pay) was pretty close to the amount of cash we admitted to having, before getting the price... It could have been worse. It did not take long. The car was undamaged, and we were soon on our way again.
By the way, there is nothing to see at the ghost town, unless perhaps you pay to go on a guided tour at the set times. Maybe they take you somewhere that is not visible from the road.