I haven't done this either, but I think that a strategy that would work for me would be to build a larger "second emergency fund" in a taxable investment account.
Build it to the size that you want your main fund to be ($6,800) and then fund it again with your existing emergency fund when it gets up to size. Then, you get two benefits. First, you have more money invested, which is good. Second, your emergency fund can take a serious hit in the market and still be where you need it to be. Even if you don't end up needing it, you have more money invested.