This question is essentially asking how to go about searching as many mid-1930s newspapers and magazines as is feasible. I don't even know where to begin. Some major papers that are still in business such as the NYTimes have digitized their archives and put them on the web, but I would guess many haven't fully digitized back that far yet, let alone put the archives up on the web.
I am interested in assembling as many press clippings about the US 1936 Olympic hockey team as possible. I have a budget of perhaps $1,000 USD. I'd be happy to pay an archivist to do the work, if that's an option. I am in DC at the moment and could conceivably access the Library of Congress.
I hope "How to research History" is a valid topic for this site.