I started looking up various competitions for composers and their new works, and surprisingly encountered numerous sites dedicated to just such a thing. Many organizations will hold competitions to promote a school, or the org, or a scholarship, or to raise money. There are also many professional ensembles out there that put out "call for scores" where they invite composers to send in their works. Sometimes winners are also awarded a cash prize. Other times not. But the point is not to win extra cash (although that's a plus). The point is to share your work and spread your name.

I'm seeing that there are lots of these calls for scores and other competitions that ask for similar instrumentation (i.e. the piano trio, string quartet, etc.). It would be useful for a young composer to learn early on how to arrange for the popular requests in instrumentation. Start writing for quartets and trios. Learn how to mimic the greats. Explore multiple styles. And do not neglect the score. Put detail and design into a score. Format your work. Make it unique. Help it flourish; it must stand out to get noticed.

While giving away the most interesting musical materials at the start of a piece may not be the most appropriate for compositional design, be aware that judges will play samples of your work from the beginning, but may not get through the entire work, so make sure the first notes heard on your recordings sound intriguing at least. Even if you don't win a competition, the judges are usually prominent people in their fields, and you have an opportunity to leave an impression on them with your work. Do your best.