Verizon is CDMA. You can set it to prefer Verizon but still use other towers IIRC. It is supposed to have the best total coverage. There are several factors which can influence this.

Phone set to exclusively use Verizon?

Does it have analog capability or is it digital only?
Analog phones can get good signals off of old towers where digtal phones can't.

Are you in a dead zone where most people don't get service?
(What technology are the local towers on?)

There are probably several other factors.

GSM coverage isn't very wide spread here based on the local coverage maps.