Apartment-hunting in Zurich is a nightmare for many foreigners and people on low incomes. I'd suggest to check out the ad-for-homes area here on the forum. Maybe you find something there.
My experience.... although I speak German, I'm still a foreigner, therefore always "secondary choice" when it comes to renting a flat.
So, my advice.....make sure to have everything together and make a good impression:
1. Always have three pages (neatly put together) with you when going for the first visit:
No1: an application letter (preferred in German) stating your contact details, birthday, origins, etc and explaining why you want exactly that flat and what a great person you are and what you do for a living, why you came to Switzerland etc. maybe even with a nice picture of you and more importantly some swiss references (boss, colleagues).
No2: a copy of the Betreibungsauszug. You get them at the Kreisburo. No need to use the originals. They are expensive and most people accept just a copy anyway.
No3: a copy of the permit. (The work contract I would only supply a copy of when I was being considered for a flat, but it was usually only needed when it came to a rental agreement)
2. Make the best possible impression. Most landlords and current tenants (who could provide a positive recommendation) here are very much about the personal touch. You gotta be well dressed, always punctual, friendly and of course have all the paperwork with you already. I personally made sure if I really wanted a flat, that I was there early and chat with them about all sorts, not mainly the flat, so they were at ease. And if there were more people at the same viewing, I made sure that I could engage the tenant or landlord most of the time. Then after filling out the application form (where it comes in handy if you have a good salary) and leaving the three pages I was talking about...... fingers crossed. But don't wait. Call the landlord or agency or tenant a day or two after to enquire about the status of your application.
Last year I did all of the above. I have applied for three flats (in the busiest rental time, which is Aug-Oct) and got offered all three. The other ones I viewed, I never applied because I didn't like them or the landlord. Of course it's more difficult in English, but friendliness, reliability and a good image are a pretty universal language.
Just think about what makes you stand out from the crowd and why they should let this flat to you.
Good Luck with your search!