In a past startup, we had at least one person apply to our regular job postings with a cover that transparently said "I know this is a full-time, long-term posting, but I really want to be a contractor for a bounded time." Since it was a great fit and they were available right away (and we urgently needed more people), we made the "hire" and ended up working together for a while. Only worked because it was quite transparent and up front in the application though.
There's plenty of comments about searching one's network, but I was looking for a comment that mentioned this. Startups do tend to prefer permanent full-timers, but hiring the right person takes time and startups also very much like getting the work done.
At one point I was unsure about joining a startup and it was them who suggested doing a temporary contract as a way to test the waters. In that case it was only a week but it was also enough for me to decide to join full-time. If joining full-time is a possibility you'd consider, I'd also mention that to the startup early on.
