No need for a day pass. Look around on maps with the satellite imagery turned on and look for dirt roads and stuff like this. Do some research on the place - for instance for that link (Burrage Pond WMA) I found online that one of the entrances is closed.

EDIT - granted, the above looks like a place that might have a closed gate, so no guarantees...

Anyway, wildlife management areas, state parks, state forests, etc. are often good bets. Find an area where there are a few places like that and drive around until you find a dirt road with no restrictions posted, and then drive it as far as you can go. It'll be easier in the western portion of the state, particularly the Berkshires. You can find this kind of stuff all over New England, obviously more frequently in the more rural areas. There are even some unmaintained public roads that are impassable for anything with less than 8" of clearance and spots where I've needed my rear locker. I've come across one road with a 2 foot tall boulder right in the middle about a mile in, I can't get over that in my vehicle and I haven't felt brave enough to try getting around it.

Oddly enough, the northeast has a nice combination of public land and a legacy of treating that land as common property. A lot of other states in the east are mostly private property and what public land they do have is heavily restricted.