Another perfect match Flemming!
I believe he entered North America via Canada, as a Jes Hansen Schmidt age 20 arrived Quebec 9 June 1873 on the ship 'Prussian' from Liverpool.
And this is probably him in Klaus Struve's website, with full birthdate
http://www.rootdigger.de/:
Schmidt, Jess Hansen * 1853 9 Sep. Accused of not showing for military service and of leaving the country without the required permit.
But he will be difficult to locate in Slesvig since those counties have no 1860-1870 censuses available.
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Michael, don't be alarmed by the "left without permit" notation:
The Slesvig area was lost to German control 1864 to 1920, and many young men left to avoid being involuntarily drafted into the German army.
You will see a very large number with that notation on Klaus' website.