I was feeling a touch ornery this a.m., and couldn't resist a little good-natured ribbing; I get such a kick out of reading Steve's line about "Once you've checked all these things and it still doesn't work...." You can't help but smile in the humor of how he leads up to it. (No offense meant, Steve; it's a great line, and good because it probably makes a frustrated homeowner stop to laugh about their problem.)
I'd imagine wireless bell are more reliable today than a few years back, but I still shy away from them; I've got so many gadgets now that it's hard to keep up with them all, and their batteries.
The biggest problem I've run into over & over with hard-wired bells, is that often whoever put it in left zero slack in the line at the door button, and often when you remove the door button the wire snaps off at the screw terminals, thus making it even shorter. When I install one on a new home, I do 2 things; leave a loop of line at the door button, and use a 4 conductor stranded wire. The stranded wire isn't brittle, like the solid usually used, and you've got a spare pair available should you need them.
As to Scott's dilemma, to add a 2nd bell, he'd need to wire from the bell, and if it's in an area finished out below, I'd look at going the long way around; taking a wire from the main floor bell up into attic space, then find a suitable wall space or chase to go to the basement.
For yours, Tom, if your entry area is recessed some from the front of the house, could you access a sidewall at the entry?
Tom, you've mentioned the talent & versatility of regulars here; seems to me you fit right in. I look forward to more collaborations in the future- David