Guaranteed to get you an offer➡
I have a desperate need to for extra headcount but am looking for a "manager tools" way to go about asking. Any ideas?
From the world I live in, and this is just one opinion, results speak. Build a biz case for headcount around the results you're achieving (hopefully exceeding). As a boss myself, I have a hard time saying no to results when my folks bring me a "big ask".
You've got to document the ways in which those extra people would provide benefit to the business, preferably adding more value than their total cost of employment (is "TCE" a recognised business acronym? It should be).
In the worst case, the ways in which your team are failing the business will already be well-known -- late deliverables, poor service to internal or external customers, and so on. In that case, the benefit is easy to quantify, the only point that needs justification is that the team really is overloaded, rather than just mismanaged or unproductive (if you have a good relationship with your boss and a good work history, that shouldn't be an issue).
Best case, you get to list some valuable things that you're currently not doing and say "these are the things that the new person would do, and here's how much it'd benefit the company if it got done". As Staedtler said, it's hard to turn those sorts of requests down when they're well-argued, because the benefit is just so obvious. You might get push-back if the budget can't accomodate it, or your boss disagrees with your value proposition, but otherwise, it should be smooth sailing.
One technique I've heard of to improve your chances of getting a req, regardless of the exact reason for the need to hire, is providing your boss with a detailed plan of the first (say) 6 months of work for a new hire (including training periods, initial work assignments, periodic review milestones above and beyond regular O3s, etc). This shows you've thought in significant detail about what needs to be done, and your boss can be confident that when you do get given the new person, they'll be as productive as possible right from day one.