> Can you name some specific companies to look at for this?
Not any
specific companies, no. I helped my friend do the job search, though. The criteria we used was "The role he was willing to take had to be in-house or majority in-house" and "Probably a support role". I also realize I wrote that wall of text
way too fast because I've slightly mixed in two different peoples' stories by accident[0] one of which had nothing to do with burnout.
To figure out the first, we started at job boards and just "went through the list looking for big companies". Often, if they outsource, the job posting will be from that company, not BigCorp, but not always. From there, at the time, it was pretty easy to confirm by just hopping on LinkedIn and looking up people who worked in similar roles at the company. If you can find a few folks listing their employer as BigCorp, in your role, and the other things match -- he applied.
He was targeting power/telecom mostly[1]. And he ended up getting offers for all but one or two of the interviews he took, which confirmed my thinking that "big companies get terrible candidates". On paper (and in real life, for that matter), my buddy was an excellent candidate and incredibly smart... just burned out. The thing of it is, last I talked to him, he is still puts in more than 40 hours (remotely, though), and he was still the team hero.
> Which route did he go, the DBA team or third-party IT org?
I haven't talked to him in about a year, but he ended up taking a gig on a sort-of DBA team (sort-of in that their responsibilities covered on-prem hardware and some DevOps work, and the job listing barely mentioned DBA work ... and it was posted as a Unix Sysadmin job[2]). He's working remotely directly for the company in that role and another who took a job at a third-party IT org (HP Enterprise or something like that), remotely (with on-prem requirements on rare occasion, etc). I know the third-party gentleman ran into some problems and, I think, is working somewhere else (similar role), now though.
[0] I do this on purpose, on occasion, because 15-or-so years ago I commented on something including an innocent story on Digg (first iteration), which turned out that I didn't have all of the information on -- and it was far less innocent than I realized -- which someone pseudonomously provided. The result was a bit of embarrassment but not much more. I took it as a warning to be more careful with stories that aren't mine.
[1] For no other reason than "I worked at a global multi-national telecom, which was a 'sort-of' tech company and we in-housed most of our IT".
[2] Just. Yeah. ... Huh?! Big corporations and job postings in IT are a whole other topic.