I don't know of a company that does this sort of work, but I know of some technologies that experts in receive these types of calls. The one that comes to mind is an ERP-focused database system. It's called "Progress" by "OpenEdge". IMHO, it's awful, but this has no hindered adoption in the slightest. I wrote Progress/4gl (their query language) often enough in a prior position to have it on my resume. Every 2 or 3 months, I'll get an email/call, asking if I could be available for short-term contracting upwards of $200/hr for Progress emergencies. I have declined all of these, because I found it soul crushing to work with in the past. However, if you could enjoy that sort of thing, that's one example of a very lucrative field to dabble in.
I've gotten a few pings along similar lines for my HighJump experience (warehouse management system, for those not acquainted). And much like OpenEdge, it's pretty... rough around the edges (at its core it's basically a runtime for a VM that's programmed with a "language" (if you can call it that) driven entirely by conditional GOTOs developed entirely in a half-baked SQL-backed IDE called "Architect"; this is paired with a DB schema from hell for all data storage, and it's filled to the brim with sprocs because even fucking T-SQL is more ergonomic than anything doable in Architect). And yet, it was actually kind of fun (in the twisted, Dwarf-Fortress-esque sense of the word) to hack on that system and abuse the hell out of it.
And for some reason warehouse managers seem to swear by it, so it still gets a decent number of new customers - meaning those customers need implementers. And since it's a giant pile of hacks, the average deployment needs a whole lot of customizations - so more implementation man-hours, and a steady stream of maintenance man-hours. Thus, I get pinged every once in awhile for some long-term contract. Too bad they didn't ping me when I was actually looking for work last year, else I probably would've accepted one for the hell of it. Still tempted to; would be an interesting side job, albeit probably soul crushing.
