How do I close my *second* round of investing when the first round VCs pass?

I already got my first round, piece of cake. We had a million dollars sitting in the bank. But now we've burned through it, and we need to raise again. I'm worried that the pressure from having a low bank account puts us in a bad negotiating position. Also the fact that our initial VCs passed (regardless of the reasons) will raise flags with our new potential VCs.

What's a positive pitch for our current situation?

pressure from having a low bank account puts us in a bad negotiating position.

That's true, you are at a disadvantage when you need investment fast.

our initial VCs passed (regardless of the reasons) will raise flags with our new potential VCs.

That's very true, and this is where your biggest problem lies. This is called "signaling" in the VC community, and it is almost a kiss of death. Please see this excellent Venture Hacks discussion about "signaling".

a positive pitch for our current situation?

You need to show traction, i.e. growth in the number of happy end users who are using your product, are paying for it, and love it. And if at all possible, go back to your existing VCs, and make them invest again. If you can't do that, at least get them to agree to personally call each potential new investor, and explain that their lack of follow on investment isn't due to lack of faith in your business.

Each round would have a different series of shares- maybe common shares in the first round, Series A in the second round, Series B in the third round, etc. Your board would also authorize up to a certain level of funds in each round, and the legal documents would say "this offering will continue until the company has raised $X million."

The real question is "should you raise a second round of funding?" If your first round investors are passing because you have, in their eyes, proven that the business they funded cannot work, then you should ask yourself whether it makes sense to raise more money to burn by simply continuing in the same direction. In many cases, a business plan changes pretty significantly between the first and second round because a good team applies the learnings from the initial phase of the business to optimize the chances of being successful going forward.

Often, this means that the second round is more of a re-start than a follow-on from the first round. Your first round investors may or may not buy into the morphed business plan, but if they have faith in you as a team and believe you are facing a big opportunity, they may decide to invest again. If they do not, it may be because the new plan is too different from the one they originally invested in and this is easily explainable to a new prospective investor.