A true type C Europlug adapter should fit in a receptacle anywhere on the European continent. Some older non-grounding adapters used to have rounded ends that could interfere with the pointed ends of the Swiss receptacles recess, but I think all newer ones will fit.
However, it depends on what device you are plugging into the adapter. Type C plugs are not grounding and are limited by codes to applications drawing less than 2½ amps (575W on 230V) so they should not be used with hair dryers. I think most hair curlers/straighteners are less than 575W so they should be OK. Electronic devices, USB chargers are usually very low wattage devices and can be used with a type C adapter.
Additionally, any American device that has three prongs (two blades and a round ground pin) should use a grounding plug adapter. Grounding plugs in Italy and Switzerland have three pins with patterns unique to that country, so if you need a ground, you should get a grounding adapter for that country. For all other countries in Europe, the French/German, type E/F grounding plug adapter, like this one, should fit in any receptacle anywhere else on the continent.