No reason you shouldn't be able to get in with Big4, you'd want to apply to their experienced hire section. Having 5 years of private experience would equate to potentially starting as a second year employee, however it can be a tough road as you'd have no auditing experience but be expected to tackle somewhat difficult sections, so starting as a first year might be better.
Apply through the experienced hire section on the firm websites, from there stalk some recruiters on LinkedIn, ie: “Deloitte Recruiter San Francisco” and maybe shoot them an email with a very quick rundown about you. “Private industry accountant with Masters and Two Sections of the CPA Passed looking to switch to public accounting.” You could also start attending your state CPA society's meeting, CalCPA here in California is like $50 for non-CPAs, AICPA also has meetings you could look into. The key is to get some contacts with real people that can help HR sort you out. Don't have to make a best friend at one of these things, but something as simple after having a good conversion with someone as, “hey I applied on the Deloitte website and am trying to break past the electronic wall, any chance you could direct me to the best person to follow up with?”
Start this process ASAP, some Big4s offer Busy season starts, ie: January
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California CPA
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