My job is relatively secure. I have tenure.
. . .
I think I'd be comfortable with 3-6 months of expenses saved as my emergency fund . . . .
I think these are the two most important things. In my mind, the biggest risks to hedge against with an emergency fund are loss of income and medical expenses. Worst case scenario, both happen at the same time. A third risk is the risk of a large unexpected (or expected but impossible to time) expense, like needing to buy a car, furnace, major appliance, etc.
If you have tenure and don't hate your job so much that you might do something stupid, then job loss risk is relatively low and you don't need as big an emergency fund. If your job has great benefits, your out of pocket medical costs are likely not going to be that great. If you have a high savings rate, you can always use your current income and credit cards to cover unexpected expenses.
In the event you have a medical emergency that is accompanied by inability to work (and therefore lost income), the game changes completely and all of your savings are on the table. Review your potential social security disability income benefits and any long-term disability insurance policies to account for this risk.
Considering all this, you seem to be a good candidate to keep a relatively small amount of cash on hand so long as you also have a "second tier" of relatively easily accessible investments such as the Betterment "Safety Net" available to draw on if you run into huge expenses.
If I were you, I would save three months of expenses as an emergency fund in cash***, and then build the second tier of emergency savings in an after-tax investment account. Once I reached a comfortable number in the second tier, I would max the tax-advantaged accounts.
*** Based on the comment about tenure, I'm assuming you are a teacher at some academic level, earning a good-but-not-great salary. The more frugal your expenses, the bigger the emergency fund may need to be in relation to your average monthly expenses to be sufficient to cover potential large expenses.