Jumpin’ creepers! A mole cricket: the European Mole Cricket Gryllotalpa gryllotalpa? iNaturalist seems to think so. Found on the NY/CT border recently.
There is a Northern Mole Cricket (Neocurtilla hexadactyla) found in Massachusetts and, belying its name, further south, but there are no Westchester Co. iNaturalist reports for it. Is this the song that Henry David Thoreau referred to:
“It is the sound associated with the declining year — and recalls the moods of that season. It is so unobtrusive yet universal a sound, so underlying the other sounds which fill the air — the song of birds, rustling of leaves, dry hopping sounds of grasshoppers, etc. — that now in my chamber I can hardly be sure whether I hear it still, it so rings in my ear.”
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A crazy blog fundraising idea.
Another crazy idea: Brooklyn Bird Club lecture tonight on stopping the slaughter of birds crossing the Mediterranean.
Backyard and Beyond 
Maybe this is what I hear next to me at nights and early morning. I love the sound. Not like a chirpy cricket, but almost a buzz to it. I am more familiar with the traditional cricket.
Some guides to c-song http://songsofinsects.com/crickets, http://entnemdept.ufl.edu/Walker/buzz/a00samples.htm
I listened, but that is not the sound I hear. Hmm, mystery not solved.
Not every crickety singer is listed in these things, unfortunately.
I saw one at Ward Pound Ridge Reservation some years ago. My group thought that it might be a larval form of something (for me, it holds the record of least attractive insect ever), so we took pictures and I sent them to a naturalist friend, who IDed it for me. I would never have guessed it was a cricket.
My friend told me that mole crickets often inhabited damp basements, but I haven’t seen that in the articles I’ve read about them.
This one was under a log, where they are regularly found on this site. Yeah, they take a bit of getting used to, sight-wise.
Oh, when I checked my pictures, I found that mine was a European mole cricket.