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For a shorter ride, try heading out west through Allschwill and play around on the French/Swiss border. Head over towards Kiffis south of the border, and then back just on the northern side. There are no massive climbs, but it gets up to just over 700m north of Roschenz on the 278.
For longer rides and bigger climbs you really have two choices. The first is to head north-east into Germany and into the Black Forest. Lots of good roads around Neuenweg. I think the best route in is via the L134 to Kandern, although we've got the train before up to Mullheim as it means you can get into the good roads sooner.
The alternative is to head north-west and do the Ballon d'Alsace or the Grand Ballon and Petit Ballon in the Vosges. Again, probably worth training it over there to Cernay and then riding from there as riding across the plain to the Vosges is uninspiring. @Cycliste and I did l'Alsacienne Cyclosportive last year, and I can thoroughly recommend that route - https://www.alsacienne.org/les-parcours/. The Grand Ballon is 1343m, so reasonably high.
Basically, if you go to the Vosges or the Black Forest you'll find it difficult to be on a road which isn't worth riding. The only problem is getting there, as the plain south of Mulhouse is a bit dull.
Just booked some cheap flights to go stay with a pal in Switzerland in June.
He stays is Basel - I'm looking for route recommendations. I know its a while away from the proper mountain passes - with a limit of around 200km, what are the best climbs/routes I can get to from Basel?