There are two well-established methods get what you want through roundabout ways. Both actually work in practical situations. What's the best way to get what you want? It depends on what you're asking for.
There are two famous psychology techniques that were recognized before they had names, and were elaborated on in countless studies after they had names. They both deal with the perfect way to ask for favors. Which one you should use, and how you should refine it, depends on what kind of favor you want to ask.
If you want a big favor, you need to use the famous foot-in-the-door technique. Getting its name from door-to-door salesmen, who needed to get their foot in the door to make their pitch, it involves getting someone to comply with a minor request in order to get them agree to a major one. The study that first zeroed in on it was conducted over the phone. Researchers called women at home during the day and asked if they would answer a few questions for a survey. A few days later, the women were called again. This time they were asked to open their home to several people, who would search their cabinets and storage spaces for around two hours. Another group of women were called with only the second request. The women who had said yes to a short survey were much more likely to agree to an hours-long search of their house by multiple strangers.
Since then, studies have been made refining the foot-in-the-door technique. It helps, it turns out, to put a little space between the first request and the second. Multiple requests, one after another, tended to lead to a knee-jerk resistance to each request. Once a person could slam down a mental door and say no, they kept saying it. If they had a little time between requests so that each one was at the start of a fresh interaction, while still informed by the interaction before, they agreed to large requests more readily. This sounds like a technique used by shady marketing firms or people looking for donation money for their kickstarter projects, but government institutions have found that it helps in public health campaigns. People who agreed to sign a petition regarding drunk driving and are subsequently asked to always call a cab when they're drunk are, afterwards, more likely to call a cab when they're drunk than people who weren't asked to sign their names. Women who filled out questions regarding health issues were more likely to, afterwards, set up gynecology appointments for cancer screening purposes than woman who were approached solely about making the appointments. Getting people to make little concessions to health and safety can help them make larger ones later.
Everything flips, though, if you are looking for a small favor. It's best to use the door-in-the-face technique. This is the reverse of the foot-in-the-door. Make a huge request, one that is certainly going to get shot down, and then ask for a little favor afterwards. People are much more likely to grant the small favor after refusing a large one.
The emotional mechanics of this method are still debated. Some argue that both people in each door-in-the-face interaction are engaging in classic bargaining behavior. Set a high price to keep the "buyer" from balking at the smaller one. Others disagree, believing that the person granting the request just wants to help, and that people are much more likely to do that after they have had their expectations readjusted due to a first, large, request. The door-in-the-face mostly works with friends and friendly acquaintances, but this doesn't give any indication of what motivates the behavior. It's true that most people want to help their friends. On the other hand, when indulging a complete stranger, the motivation to do them any favor would be completely altruistic. With friends, the interaction could be seen as "bargaining" for the continued friendly relationship. Most people, when shown interactions that involve the door-in-the-face technique, believe that it works because of a desire to help. Then again, most people might not believe that the best way to ask for a small favor is to ask for a large favor first.
The door-in-the-face technique generally is used, or at least studied, by businesses. It was noted that it is the reverse of the foot-in-the-door technique not just in mechanics but in timing. One study, done in restaurants, tracked the conditions under which people ordered coffee after meals. The servers were instructed to offer people a chance at dessert. When they were turned down, they were to ask if the customers coffee, instead. While delay can help in the foot-in-the-door, door-in-the-face needs immediate action. Many more customers said yes to coffee if they were asked right after they said no to dessert. When the server waited another couple of minutes before coming back to ask the question, more customers said no. This might be something to take note of during your next night out. Is your restaurant using the door-in-the face technique?
[Via Personality and Social Psychology Bulletin, Health Communication, The Journal of Social Psychology, International Journal of Hospitality Management, Psychological Reports, Journal of Language and Social Psychology]