I think I can help. Short answer is it depends on your township and how quickly they reassess your new home.
Our experience:
We purchased our new construction home in PA and closed early Oct. 2012. We received a knock on the door from the Township within a couple of weeks. She did not ask to come inside, but I would not have let her inside (I don't believe you are required to let them in.). She asked a couple of questions about the home (if we had any cathedral ceilings, if the 2nd floor went into the garage, etc.) and then asked if she could take a look outside the home and take measurements (to find square footage). I agreed. I figured I wanted an accurate assessment, not a guess of square footage. (I figured they would guess bigger not smaller.)
About a month later we received a letter with our new assessment. Within a few days we got our tax bill. Our Township is on a calendar year, school district is a fiscal year--July through June. The interim bill was for 2 months (Nov. and Dec.) of Twp. Taxes and 8 months of school taxes (Nov.-June). It was due in Jan. or Feb. 2013.
We then got our regular Township tax bill a few months later which was due in June 2013. AND our school tax bill which is due in Sept. 2013. We will have paid a LOT of money in property taxes this year so have a lot to deduct! (Only silver lining I can think of!).
We don't escrow our taxes either. I estimated how much $$ to transfer to a special savings account monthly based on the initial assessment, but since we now have a year of "real" tax bills, I have recently adjusted this amount so we don't fall short. The good part of the staggered payment schedule is the account can build back up again before the next bill is due.
Enjoy your new home! I hope I helped. That first year is really a doozy!!! PA is the only place we've lived that does these interim bills. The (4) other places we've lived and had new construction we just received a bill that said our new assessment and what the tax rate would be for the NEXT tax year. Gotta love PA!

. We do like it here, though! To a certain extent, you get what you pay for!