According to The Word Detective get off the dime is actually older in origin than "stop on a dime", it dates back to the 30's during the days of the dance halls:
- Since a dime is a small unit of money and fairly easily to come by, this small coin has played a much larger role in US slang than, for instance, the hundred-dollar bill.
- To “drop a dime on someone,” for instance, means to inform on them, usually by tipping off the police, and originated back in the 1960s when a call from a public telephone cost ten cents. “Dime” has also found a home in the slang of drug users, where a “dime” or “dime bag” has long meant ten dollars worth of a drug. The small size of a dime has also been used as a metaphor for “a small spot,” as in “stop on a dime” or “turn on a dime” when speaking of motor vehicles (or politicians).
- “Get off the dime” dates back to the days of dance halls and “taxi dancers,” women employed by the halls to dance with strangers, usually for ten cents per dance (a grim occupation immortalized in the 1930 Rodgers and Hart song “Ten Cents a Dance”).
- A contemporary account, published in 1925, explains the phrase: “Sometimes a … [dancing] couple would … scarcely move from one spot. Then the floor manager would cry ‘Git off dat dime!'” Similarly, “dancing on the dime” meant to dance very closely with very little movement, behavior that might well attract the attention of the Vice Squad and get the hall closed. Thus “get off the dime” referred both to the the customer as the “dime” he had paid and to the small spot (“dime”) on the floor where the couple seemed frozen.