The guys at NJO named this weapon I wondered about – the trumpet-barrel gun used by the Pilgrims and perhaps Puritans:

The blunderbuss was in use in the 17th century, and is the weapon most commonly pictured in the arms of the Pilgrims….. The funnel-shaped barrel (either round or elliptical) is not designed to enhance the ballistics of the weapon, but serves to facilitate loading ammunition into the muzzle. This makes it much easier to refill a blunderbuss with shot in situations where this would not normally be possible (as when riding shotgun on a stagecoach speeding down a bumpy road).

From the same site where I got this passage from I read about the origin of the word "Blunderbuss" as a non-arms word of slight:

By its nature, the blunderbuss is not a very precise weapon. Although it was sometimes used in a military setting, it was more effective when the goal was not to hit a specific target, but rather any one of multiple targets, such as for crowd control. During combat, the blunderbuss was a very unpredictable weapon; it could hit an entire group of enemy soldiers or miss all of them. The blunderbuss thus became a byword for inaccurate marksmanship in any field. As well, the word itself came to mean “a clumsy or stupid person”.

I assumed that pretty much all guns pre-1700s were relatively inaccurrate, but perhaps the case is even moreso with the Blunderbuss.  Interesting stuff.