Rental vacancy rates are low, so you may be able to find an area/town you like in September, but anything available for rent in September will most likely be leased for October or Nov. 1st. Renters give notice (usually a month) and the unit gets listed for rent. The landlord wants to get it occupied as soon as possible, so allowing for a little clean-up time in between, there is usually not much more than a 6 week lag.
You didn't say where you lived in Europe, but I spent 7 years in Brussels and the rental market there was very different from here. It seemed most people rented there, while more people buy here. So there are fewer available places to rent as a result. Rental units make up about 20% of the housing supply in Winchester.
Another town you might look at is Belmont. Similar size, excellent schools, close to Boston, etc, but rental units make up close to 40% of the housing there. Parts of Cambridge might work for you too!
You should be able to find something nice in your price range. Are you looking for a single family house vs a condo? How about a townhouse?
Anyway, feel free to email me with questions.