Clubs are good options for volunteering. You can work on their Board, volunteer special skills like web design or editing, organize a particular event or get involved with their international arms which can reach as far as the UN in some cases.
If you're American or have ever lived in the States, the American Club of Zurich: http://www.acz.ch/
If you identify as a woman and have ever lived or worked in the States or have a spouse who has, then the American Women's Club of Zurich: http://awczurich.org/
If you identify as a woman, then the International Women's Club of Zurich: http://www.ziwa.com/
Unfortunately they all involve member dues, but they connect you in with an English speaking network that will certainly have valuable professional ties. They may all seem socially-oriented at first, but there are tons of volunteer opportunities lurking behind that. What's more, if they really need your skill, they may offer to have an established member pay your dues to incentivize you to work with them. Also, AWCZ has a page where they collect volunteer opportunities that don't require great German-speaking skills. You don't have to be from the US or a woman to contact them; they're glad to help anyone.
List of volunteer opportunities from AWCZ: http://awczurich.org/about/phasepage