Hi dhavlena,
I've heard the exact same peeping sound under the same circumstances you've described. A bird that sounds like a Spring Peeper, but calls from high in a tree at night for hours on end, and hops from tree to tree. It's quite loud, and can be heard through closed windows. (I've never seen it, though.)
It's definitely NOT a Spring Peeper, because it calls from September to April, and the Spring Peepers around here (Connecticut) *only* call in the spring (early March to late May), and (from what I've heard anyway), they call from low brushy areas at or near the sides of ponds or small lakes, not up in the trees. The peeper I've heard is calling from the very tops of the trees (oaks and maples) on our mostly wooded property.
Spring Peepers also tend to quit their peeping around midnight, while I've heard our mystery tree-top peeper at 2:00AM and 4:00AM. And Spring Peepers usually peep in a large chorus. It's rare to hear only one peeping at a time; that usually happens only for a short while when the temperature is a bit chilly for all but one extra-hardy frog.
The best bet for an ID I've seen so far is the Swainson's Thrush, which makes a *somewhat* similar call while flying overhead at night during its migrations. And several people I've seen online have described the Swainson's as "sounding like a Spring Peeper." But I've listened to several Swainson's Thrush flight-call recordings, and I think it's a bit different: softer, breathier, thinner and sometimes more of a "wep" than a "peep." Not as "urgent," as you say (good word choice), either. And Swainson's Thrushes reportedly migrate in huge flocks, so that you can hear hundreds of them peeping overhead all night long. What I've (and it sounds like you've) heard is a single bird that stays in one general area, at most hopping from tree to tree. Perhaps what we've heard is a louder variation of the flight call, when the thrush is resting in a treetop before continuing the migration?
I listened to the 8th Flying Squirrel recording and thought it sounded pretty different.
But I'm not sure about that Swainson's Thrush ID. It's definitely been driving me nuts that I haven't been able to solve this mystery! I'll keep looking and post if I find anything interesting.