Elms for USDA zone 5 and zone 4 - these species should be able to be wintered, outdoors, without temperature protection, pot set on ground, protected from wind & maybe sun, mulching in optional.
Ulmus americana - American elm, in the landscape susceptible to Dutch elm disease, in a pot treatment is relatively easy. Leaves will reduce dramatically, almost as much as Chinese elm. Native to most of North America.
Ulmus thomasii - rock elm, also known as cedar elm, but it is not the same species called cedar elm in the southern USA. Native to northern Midwest America, into Canada.
Ulmus parvifolia var coreana - the Korean ecotype of the Chinese elm. Chinese elm from southern China may, or may not be winter hardy in zone 5, the Korean ecotype has survived through -29 F (-34 C) in trials at the Morton Arboretum in the west suburbs of Chicago. The Chinese elms I have tried, of unknown origin did not do particularly well. Worth experimenting with different sources.
Ulmus pumila - the Siberian elm - superior winter hardiness, an invasive species, to keep lower branches from weakening and dropping off, absolutely must have full sun pretty much from sunrise to sunset. If given enough sun, it works well for bonsai.
Ulmus glabra - the wych elm - this is the most northern, cold tolerant of the European elms - there is a tree or two surviving in Greenland. If you can find it, it can work in zone 4.
Ulmus rubra - native to the northern midwest NA into Canada, the Slippery elm, or red elm - looks like an American elm except "chiefly distinguished from American elm by its downy twigs, chestnut brown or reddish hairy buds, and slimy red inner bark" - flowers in spring before leaves appear, and flowers are reddish. This elm is a little bit more shade tolerant than some other elms. Still, full sun is usually best for cultivation.
Ulmus minor - the Field Elm - a widespread European elm that is very cold hardy. Possibly invasive in some areas of USA.
So the above are either native to the USA or somewhat available in commerce. The few below you are very unlikely to encounter, they might be hardy.
Rare in North America elms that might work - Ulmus davidiana, Ulmus glaucescens, Ulmus macrocarpa, Ulmus bergmanniana, Ulmus lamellosa,
This list is only a complete as the information from Wikipedia. There might be some trees I omitted. Apparently someone from the elm evaluation and breeding program at Morton Arboretum, in the western suburbs of Chicago had edited in some good information into Wikipedia.