IntroComp is one of the coolest low-key events on the IF calendar that hopefully will get a lot more attention this year with a snazzy new website and new management. This is the one ongoing competition that actively solicits unfinished IF works. It's a great way for new IF authors to get their feet wet if they've never finished a game and attempted a comp, or want some support and motivation to do so.
One major change this year is IntroComp is accepting any slice of an unfinished game as opposed to just the beginning; the website makes clear the "Intro" part of IntroComp now means you are "Introducing your game" as opposed to providing just the actual "intro" as required in years past. The website rules state authors may submit the middle or end of a game as well. There are also cash prizes with a catch: the author has to actually finish the game within one year and notify the Comp organizers to claim a prize.
Even if a submitted game doesn't win Introcomp, one of the coolest parts is voters are encouraged to provide constructive feedback along with their votes which is provided to the authors.
The only slight downside is works submitted to IntroComp are ineligible for IFComp, due to being a partially published work. However, Spring Thing is a good venue to show off a game that was partially developed via an IntroComp entry.
If you've got a long-unfinished game fragment kicking around, perhaps consider submitting it to IntroComp.
Intents to enter this year are due by June 30, and the "complete" game slice for the comp must be submitted by July 31.
See http://introcomp.org/ for more details.