If you are offered a set period of time, for example 6 months or 2 weeks at a given company, either as a freelancer, or on a graduate scheme, I think you need to establish very very quickly how you can offer the company something no-one else on their books can. Observation is key; check out who works there, see what they do well. Have they got all their bases covered, and are there things they do better than others? Once you've established these 'gaps in the market' - pounce on them. Maybe you can do it faster. Maybe you can do it quicker. Maybe you can come up with alternative ideas. Maybe you're the biggest contributor in an ideas meeting. The only option I would definitely steer clear of is - maybe I can do it cheaper. No one wants the reputation for being the cheapest, and companies don't want that kind of boast in their corner.
So, observe, assess and adapt accordingly. It may sound a bit OTT, but I guarantee it will do you no harm to get into this mindset. Remember, if you want a job, and you're given a foot in the door, it's up to you to show why you deserve to be kept on. Don't waste an opportunity.
Me, I'm still finding my niche here, there is a lot of talent at Dagobert, but I'm sure I can establish myself in some way. I'll keep looking.