From my experience, many start-up companies find new clients and win contracts via various freelancing platforms; these are not just for individual contractors. I've worked with companies helping them get started on freelancing websites and win clients. It takes time and effort to start, but once you have established reputation you get invited for jobs and re-hired. When using freelancing platforms your payment is secured which is important for start-up - not all sites offer this though, always read the small print carefully.
You should also network locally. Find an event that is geared towards start-ups and surely someone there needs design work done.
If you have any further questions, just give a call.