What is the southernmost state in the United States?

I'd say Hawaii.
https://en.wikipedia.org/wiki/Extreme_points_of_the_United_States

Continental?

Key West Florida.

"If you exclude Alaska and Hawaii, the answers are:

Most Northern Point: Angle, Minnesota, at 49º 23'N

Most Southern Point: East Cape, Florida, at 25º 7'N

Most Eastern Point: West Quoddy Head, Maine, at 66º 57'W

Most Western Point: Cape Alva, Washington, at 124º 44'W

http://www.worldatlas.com/aatlas/infopage/nsewusa.htm

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I say Key West only because it is the "recognized" SMP in America.

http://www.roadsideamerica.com/attract/FLKEYsouthernmost.html