You usually get one shot at this, not a back-and-forth. That changes everything about how the ask needs to be built before you ever open your mouth.
Asking your own manager for something benefits from context they already have, they've seen the work, they know the backstory, they can ask follow-up questions later if they need to. An executive several levels up almost never has that context, and rarely has the time to build it in the moment. If your ask depends on them understanding a long history first, you've already lost the room before you've made your case.
An executive isn't evaluating your problem. They're evaluating whether saying yes to you is a good use of their scarcest resource, attention.
The version that works compresses the ask into something that can be evaluated in the time it takes to read three sentences: what you need, why it matters to something they already care about, and what happens if they say yes versus no. Everything else, the backstory, the alternatives you considered, the full analysis, belongs behind that, ready if they ask, not in front of it.
"I'd like your approval to add one contractor to the migration project for six weeks, at roughly $18k. Without it, we're on track to miss the Q3 deadline you flagged as a priority in the all-hands. With it, we hit that date with margin. Happy to send the full breakdown if useful."
Notice the ask leads with a number and a timeframe, not a build-up. It ties directly to something the executive already said mattered, rather than asking them to take your word for why it's important. And it closes with an offer of more detail instead of dumping it all up front, that offer respects their time, and most executives will either take you up on it or trust you enough not to need it.
Timing and setting matter more with executives than with almost anyone else you ask things of. A hallway ask, a rushed Slack message, or the tail end of an unrelated meeting rarely gets you a real answer, it gets you a "let me think about it" that quietly becomes a no by default. If the ask is significant, request five specific minutes for it, state clearly what you need those minutes for, and come with the compressed version ready the moment you get them.
One more thing worth preparing: the version of the ask you'd make if the first answer is no. Executives say no far more often to an open-ended ask than to a specific fallback, "if the full budget isn't available, would a smaller three-week pilot work?" gives them a path to a partial yes instead of forcing an all-or-nothing decision on the spot.