Enthusiasts of frequent-flier mileage have all kinds of crazy strategies for racking up credits, but few have been as quick and easy as turning coins into miles.

At least several hundred mile-junkies discovered that a free shipping offer on presidential and Native American $1 coins, sold at face value by the U.S. Mint, amounted to printing free frequent-flier miles. Mileage lovers ordered more than $1 million in coins until the Mint started identifying them and cutting them off.

Coin...