I personally don't get emergency funds, in a pure savings account sense. I would argue you'd be better sticking all of that in with the rest of your taxable investments and start earning closer to 5%, if not more. If an emergency comes up, you can quickly sell and transfer, usually in under a business week.
Doing some back of the napkin math, if you put $25k in some sort of index fund and it got 5%, in one year you'd earn $1250. If for some reason, you had to sell it all off and pay the taxes on the gains, you'd be paying less than $400 in taxes, still netting you $800, which is way better than the $12 or $175 you're getting in a savings account.
I'm sure my math is way off but the general principle should hold true. If you're worried about a week being too long, wouldn't a couple grand suffice until the rest could be sold and transferred? I would keep a couple extra in my checking, kill the savings, and invest the rest.