A quick heads-up before diving into the story: I've set aside for you a complete shopping list of every ingredient you'll need so you can recreate this classic at home. Here's the tale.
The Between The Sheets is often attributed to Harry MacElhone of Harry's New York Bar in Paris, who is said to have invented it in the 1930s. The drink is essentially a Sidecar with the addition of white rum, and is usually traced to the brothels of Paris, where maîtresses allegedly offered it to clients as an inviting welcome, whispering its suggestive English name.
A parallel theory places the origin even earlier, in 1921, at the Polly's Apizza in New York, where a French-born bartender would have created it to boast a drink "strong enough to put you between the sheets." True or not, the cocktail belongs to the naughty, tongue-in-cheek cocktail culture of the Roaring Twenties, and remains on the IBA list of official classics.
To recreate this one with the right style, tools matter as much as the recipe. I use and recommend a complete barman kit like this set on Amazon, which covers every essential tool at a very reasonable price. For a more serious setup, a professional cocktail bar station keeps bottles, glasses and tools organized and close at hand, instantly upgrading the whole experience.