How do you advance through mercenary tiers in AC:Odyssey?

I've been playing AC Odyssey for a while now, and haven't figured out the trick to advancing through the mercenary tiers. For example, my position/rank is in the middle of a tier, so I mark the first mercenary in the next (lower numbered) tier. I track them down and defeat them, only to have my tier unchanged. The defeated mercenary now appears in my same tier, to my immediate left (in a lower-rank position).

This has happened several times. Eventually, I defeat a mercenary and my tier advances (lowers). I haven't paid attention to what might have been special that advanced me. What am I missing?

You can only climb the ranks one place at a time.

When you defeat a mercenary who is better ranked than you are, you only advance by one position in the overall list. It doesn't matter which tier the other mercenary was in, only that they're higher ranked than you - besides that, the difference is irrelevant. Most tiers have 5 mercenaries in them, so most of the time to advance from the beginning of one tier to the next will require you to defeat 5 stronger mercenaries; there's no way to "skip ahead" and progress through the tiers any faster.

As the Ubisoft Odyssey FAQ states (emphasis mine):

Defeating other Misthios will not only earn you rewards, but also recognition - every time you take down or recruit a mercenary higher up in the hierarchy, you will move up in your respective tier.

[...]

To reach the top tier, you must defeat 52 higher ranking mercenaries.

As you can discover higher-ranked mercenaries in the world simply by stumbling across them or attracting their attention by earning a bounty, you'll eventually advance anyway if you keep defeating every mercenary you encounter.