Even after a year of underemployment, I didn't access my emergency fund. And even then, if something were going to be a financial "emergency," 99.9% of the time I would know it about a month ahead of time.
The only close call I had was when ING incorrectly processed my rent check and bounced it back to my LL. The LL wanted a cashier's check immediately and I didn't have a brick and mortar bank to get it from, but of course an eviction takes a while so the LL didn't have much of a choice than to accept my assurance that I was putting a paper check in the mail that moment, that ING acknowledged that it fucked up, and that I was paying the bounced check fee (which ING reimbursed me for).
Otherwise, any other big expense that comes down the road can be paid for with my credit card, and then I have 30 days + to figure out how to pay that.
The only other scenario that comes to mind, where you would need $$ immediately, is if you need to post bail. What's the likelihood of that?!
All that is to say that you should put your E Fund wherever it will earn the most $$ in your risk comfort level. If it takes a while to get at the money, all the better for your psychological obstacle to using that money for anything less than a true emergency.