I don't think there's a tried and true way to do it--but I can tell you what's worked for friends of mine/myself in the past.
I'd sent out cold resumes like you had and never heard a thing-but low and behold a director I worked with has a Broadway PSM he loves to death and suggested we have dinner. Seemed totally awkward to me to have a networking dinner with a stranger, but I gave it a shot.
We had dinner, got along great, he asked for my resume and etc and a month later he offered me a PA position on a Broadway show. Now, unfortunately the shows budget ended up not being flexible enough to afford PAs for the rehearsal process but I still have that contact who I follow up with on a regular basis.
So it really can be who you know and networking... so start chatting everyone up!