Non-metric? Who cares? There is no name for it.
Since until 1970 the metric system was meaningless in the US, there was no need for a name for the system in use. It was just the standard. Some of it is based on the imperial system, but if you actually *used* the imperial system, you'd get a lot of things wrong.
Also, since there is no confusion in unit names between the two systems, there is still no need (for the most part) to distinguish the metric set of measurements from the non-metric ones. The only exception is the ton, and if someone in the US is referring to those, they say "metric ton". A US ton is, get this, called simply "a ton". Nobody ever, ever, says just "ton" and mean "metric ton". *