I've most definitely experienced this issue first hand a few times, and I struggled to find an answer. I finally did get a sense of how to tackle this by going out and meeting with founders of engineering and other professional service firms. The product is quite different (not to mention billable rates...) however, the model is very much the same.

I would always avoid taking financial risk to fund growth, unless you have a contract in the pipe to cover the salaries, overhead and even the profit margin all lined up, in which you really really need to hire correctly and be seriously organized.

As a rule of thumb, every billable resource has be billing at 85% of their work day, everyday. This is incredibly hard to achieve when setting up your agency, as there are countless processes that need to go in place to achieve this. Your first 5 or so hires all need to be billing at 85% before you start taking on overhead staff. Taking on overhead staff too early WILL limit your ability to scale.

If you have a team all billing appropriately , it becomes quite easy just to have two staff working the extra 15-20 hours per week to cover the additional outputs you require. Only make the hire once you have the business in place, solid internal processes figured out ahead of time and have gone through a billing cycle or two so the additional salary won't eat up your cash flow.

It really all comes down to getting your model down to an art. It is more often far easier to work with what you have to become more profitable than to scale. And if you are looking to scale before you have at least 3 months cash in the bank, I'd be very cautious about growth.

This was the stuff I never wanted to hear, but really really needed to hear.

Also, if we are talking about your first hire, first look at ways to offload a lot of your overhead time suckers. (Invoicing, book keeping). If you are billing enough and charging a fair rate, it won't be long before you have the cash in the bank to handle the risk. If not, you are leaking money somewhere. Find the hole and plug it, fast. Before you make your first hire, I recommend reading "The E myth". This will help prepare you to ward off some of the perils of your first hire.