I'm afraid this is part of the price you pay for being a contractor. There are many benefits to being a contractor, as I'm sure you know. But, in return, you have to live with the problem of not being able to look for work until you are ready and available to take it on.
This isn't just to do with switching from salaried employment to your first contracting job. You'll find the same things happens at the end of every contract. You won't be able to look for new work while you're still working on the current contract, because the new client will probably not be prepared to wait (well, some of them might, but probably not for more than a week or two at most). And once you do start looking for new work, you'll need to find it quickly otherwise you'll be without any income.
This is one of the reasons I have never done contract work myself. But I have several friends who are contractors, and they all mention the same problem. On the other hand, they all enjoy contract work, and would probably not want to go back to salaried employment.