Hi there,
First, never directly ask for a referral! Rather, hope they offer, and if they don't, you can "hint" at wanting one by saying you're interested in the company and will be applying soon.
Second, you do not need a Partner for a referral to be effective. Even Consultant referrals matter a lot.
Third, if you've had dozens of calls and still are not getting a single referral, you're doing something wrong. You need to revise your strategy, and possibly hire a coach to help you.
Fourth, this process is critical to improving your odds of getting your resume selected. 80-90% of all jobs are now obtained through some sort of connection/network (i.e. not a cold resume dump). A referral puts your name at the top of the pile. Lacking a referral, even someone forwarding your resume to HR helps increase your odds. Finally, networking with a few people in your target office increases the odds that they mention something to HR AND allows you put their names on your cover letter.
In terms of networking tactics:
a) Reach out to people in your network
b) Reach out to people once removed from your network
c) Reach out to people with a similar backgorund to you (i.e. same alma mater, same hometown, same career switch, etc.)
d) Tailor a message to them specifically both showing interest in them and their journey and demonstrating that you have done your research and could be a valuable hire
e) Play "tag" across calls you get so that you can work your way towards the company/office/role you want
f) Leave a strong impression - i.e. don't ask "basic questions". Rather, demonstrate your ability/intelligence by asking smart questions, carrying a natural conversation, showing genuine interest, etc.
g) Never directly ask for a referral, but "hint" at needing one (this is nuanced and important...happy to talk through wording)