Say you saw a flat tomorrow and like it. You could probably give the deposit the next day, and agree for the move in date to be a week's time, the 11th.
If that happens, you'll be paying rents for the three weeks overlap. I don't know if you can afford that. I'm a worrier so I was happy to absorb that cost for peace of mind when I was in a similar situation.
If it takes you longer than a day to find a place (which it will) and/or if you can get the new landlord to give you a longer period of time than a week, then the overlap will be shorter.