[EDIT: This article was poorly researched, and apparently there is some discrepency between the rules as they are in the draft and how they will be in the final rules. I'll revisit this post when the final rules come out, for now just skip the section called "Tug of War" and you should be good. ]
When it comes to task resolution in FATE, Contests are kinda the middle child. Challenges are a single opposed roll and are easily used to resolve a problem, and Conflicts are meaty. Because of this, I think contests don't get much attention. That shouldn't be the case, so here are a few ideas for how to hack contests.
Tug of War
By default, each side tallies their victories, and the first to get past three wins. When both sides are competing to see who achieves an outside goal first, that's fine. But sometimes you're directly competing with each other over a single goal which only one of you can achieve. The simplest example of this would be two teams playing Tug of war. Another would be an open-ended chase, where all that matters is if the pursuer can close the distance, or if the hunted escape out of sight.
In this case, you could set the victory as a slider between -3 and +3. Victory moves you one notch in the direction towards your goal. Getting to your end means you've won the contest. Gains by you directly offset gains by your opponent. Another similar concept is to keep the two separate victory tracks, but to allow you to use a victory to remove one of your opponent's tally marks instead of add one to your tally
This can fairly model a number of situations better than the standard contest, but it has a rather serious fault: Two evenly matched opponents will tend to move the victory slider back and forth with neither one able to get it all the way to victory. For this reason contests like this can drag on. I would only use this option if I absolutely could not frame the situation so that the default first past the post scoring worked.
Moving the Goalposts
Anytime I see a constant in a game design, my first reaction is "why isn't this a variable?" Often there are good reasons, but just as often you can get some mileage from messing with the values.
In this case, some tasks require more work to accomplish. In a chase, for example, getting to a safe haven 50 miles away is harder than one only a few blocks away. It's a lot easier to convince a few friends of your side of the story than, say, a judge and jury. In these cases you might require more or less than the standard 3 victories to win the contest.
Adjusting this number pretty directly effects how long the contest will go on for. Dropping the number down to 2 will speed things up, and increasing it past 3 will drag things out. But more turns gives the players more time to do interesting things, story-wise. Entire movies involving courtroom Drama might be played out as a single contest with a much larger victory track.
It gets even more interesting if one party has a longer victory track than the other. Lets go back to the chase example: The hunted is trying to get to their safe house several miles away, while the pursuer is only a block behind them and only needs to close that distance. So the hunted might need something like 10 victory tallies to get away, while the pursuer might have only to achieve three. The hunted better have some strong advantage!
The victory track is, as stated above, just a pacing mechanism. That's also what a stress track is. Perhaps one can serve as the other? In this case, your goal has a stress track, and your actions to achieve your goal become attacks meant to overflow that stress track. By default that track would have three boxes.
This has some speeding up effect, since multiple shifts do more stress. It also unifies all pacing mechanisms under a single mechanic, which some of us gearheads find more elegant. But the main thing this accomplishes is removing this difference between a contest and a conflict. Everything is a conflict, and your goal simply becomes a valid target on your turn. This means that all the tactics and complexity that makes conflicts meaty can be brought to bear on contests as well. This is great in those situations where your hero is directly conflicting with someone while still attempting to reach a goal. Punching each other while also trying to climb a broken rope bridge, for example, or perhaps a running gun battle through a collapsing building (with the aspect "On Fire", or course), while trying to be the first to reach the only exit to safety.
Of these three options, I'm most enamored with Moving the Goalpost. It's simple, easy, and allows the GM to model a wide variety of situations. I have several ideas for Apocalithic that are going to rely on this to implement. The others are ideas that I don't foresee getting a lot of mileage out of, but I'm sharing anyway because it might give others something to think about.