There is no move. However, there are mechanics.
The mechanics are "you, the GM, make something up". This may sound shallow and cheaty, but it's not; you're supposed to be writing new mechanics all the time. As the plot moves beyond the generic pressures of a first session and into more specific territory, you're supposed to be developing mechanics to cover the specific things you and your players have become interested in.
Note - you and your players. You can't force your players to be interested in anything, it'll seem super fake and weird if you try. But if they hired people on and you named them and gave them life the way you were supposed to, maybe they've got attached. Or maybe you sold them on a campaign about people adventuring in a far frontier, who don't know much about the local customs and need to rely on hired help more for navigating the civilized parts of the frontier then the dungeons (but they're useful in the dungeons too). In either case, somewhere down the road a piece of narration like this might strike your fancy:
A forge complex has grown to dominate this town. However it happened, the immediate upshot is fairly obvious. These are the best- and most-conspicuously-armed guards you've ever seen, and quite a few of the more rough-and-ready citizens are walking around with a certain deliberance to their stride that suggests they've got some heavy armor under their cloaks and clothes.
Regdar, whose patched chain shirt was shiny and new when you hired him on, picks at it self-consciously before turning to you. "Hey boss. You feeling a bit underdressed here, or is it just me?"
Your extensive experience allows you to translate from Regdar-speak: "please please please can I have a pay advance to buy some platemail".
And assuming your PCs humored him, it'd be really weird if Regdar got an expensive new suit of armor but that didn't make him a better protector or warrior at all. So maybe you spell that out too when you tell them the requirements or consequences and then ask.
This is a pretty big chunk of change Regdar's asking you for - plate mail goes for 350 coin. It'll absolutely settle up your books with Regdar, now and for the next let's say three sessions? I'll make some little "paid up" circles in my notes here. It'll also give him three more points in Protector, and one more in Warrior since he'll be a little more willing to jump into danger.
But... it's 350 coin. You're not going to starve any time soon but it's a big chunk out of your surplus, and Regdar knows that too. You're not going to take a Loyalty hit if you just keep paying him regular-style, but making this big spend will be an option as long as you're operating in this neck of the woods and the forges are fired.
So, do you take him up on it now? Never? Or do you want to wait and see?
An Interlude: Bait Is Not Hook
So, just to dissolve all of the plot off that, the GM is making the move tell them the requirements or consequences and ask. IF you spend 350 coin, THEN you pay Regdar for 4 sessions (including this one) and he gets +3 Protector and +1 Warrior. "Hard" and "soft" moves are kiiiiind of a fake idea, but as this involves no permanent changes to the players or world state it's a "soft" move.
When the GM then increases Regdar's skill ratings after the players spend 350 coin, that's not a move at all. That's just basic GMing, you know, describing the world, saying what honesty demands. You can tell it's not a move because it doesn't make any demands of the players, which is what moves are for in the GM's hands - to wrap their piece of the conversation and give the players a push to start part of theirs. What does "Regdar now has +3 Protector" demand of the players? Maybe a few seconds of bookkeeping?
I guess I'm saying - don't just think of moves when it comes to things you're going to say as the GM. You have a raft of principles to uphold and in addition to moves, there's rules, prep, and descriptions in your how-to to get it done.
Planning for Upgrades
One of the things you might have noticed about hirelings is that they have more starting points to allocate depending on the size of the town you hired them in the first place. Part of this is likely because more skilled people are going to congregate in the bigger towns because that's where their prospective employers have the deepest pockets, but part of it can also be that as a town grows it generally has access to higher-quality equipment. There are a couple things to note about this assumption that might get in your way, however.
Hireling skills are equipment-dependent to varying degrees. It's easy to imagine buying better weapons or armor to improve a hireling's combat skills; after all, haven't some of the PCs done the same? It's a little harder to imagine the equipment purchase that would make someone a better Adept or a better Tracker.
Hireling equipment isn't a preset menu. What does buying platemail do for Sackville the Burglar? They're not a combatant as is and buying some armor for them isn't going to change that. Are they even going to think you've done something valuable for them at all? It's better to think of the platemail as an opportunity for Regdar; opportunities for other hirelings may happen elsewhere in town with different costs, or they might arise in the wild - Sackville recognizes the work of a wandering or forgotten locksmith, and demands to test their skills against it, which is going to mean that you all protecting them from whatever happens until they've finished.
If you don't think it's practical or a good use of your time to integrate bespoke hireling skill upgrades into the fabric of your original adventure, you might want to try something like this out, replacing your own planning and judgement with a more universal rider to a move that already exists:
When you release one of your hirelings from your employ to train, re-equip, or otherwise invest in themselves, note down their final-loyalty. The GM will tell you what will happen and how long it will take. At the end of this training period, you can attempt to hire them back on; take +final-loyalty on your Recruit roll, in addition to the other modifiers.
At the GM's option they may be there on a 7-9 or 6-, with their training having gone off-course or failed disastrously. On a 10+ they are definitely there and will ask to be re-hired; however, turning them down will incur the -1 forward penalty on Recruiting for turning down applicants.