I am in my seventh year after my PhD, and am applying from a tenure-track position in Mathematics. I am working with a job coach, who is suggesting a 2.5 page cover letter for tenure-track mathematics positions. This seems long to me, I always thought that in math 1 page was max, but are the rules different when applying mid-career?
I think that the length of the cover letter is the type of thing that differs greatly between research institutions and small liberal arts colleges.
At the small liberal arts college where I work, 1-2 page long cover letters seem to be pretty common. (I'm also in math.) There's no length requirement of course, but it's expected that in addition to the normal stuff (who you are, a bit about your teaching and research experience, potentially stuff about working with undergrads if it is relevant) applicants say a bit about what makes them a good fit for the school. Basically, small liberal arts colleges tend to have their own distinctive personalities, and it can be a major problem if your cover letter comes across as super generic. Also, remember that a lot of small schools don't really have research groups in any particular area. Thus, if they're hiring an analyst one year there is an excellent chance that it's because the previous analyst just left / retired. People will read your research statement of course, but many won't make it past the first page or two. In contrast, every member of the department will read your cover letter. So if the cover letter you send to our small liberal arts college looks like you could have sent it to a large state school as well, that'd be an issue. Having said all of this, it's not like you need to be super long winded in your cover letter. If you can say everything that you need to in less than a page that's totally fine! It's just that a lot of people will be reading your cover letter, so if you want to explain why you're committed to working at a small school like ours or explain the context of your decision to apply to our school (which could very well be the case if you're applying mid-career), this would be the place to do so.
I don't have any direct experience with being on a hiring committee at a research department, but my impression has always been that the cover letter is considerably less important at these sort of places. (Perhaps it's even seen as an unnecessary formality? I've always found it weird to read 'Enclosed you will find...' when it's just a PDF file on MathJobs and not even a physical letter.) For example, when I was a postdoc I had a number of friends that used a computer script to insert the names of the schools they were applying to into a generic cover letter. It's not clear to me that this affected any of their searches in the least. Hopefully someone with some experience at these sort of places will respond to your question, but I think that at research schools there are much more likely to be well-defined research groups with members that are going to be able to read your research statement carefully and already know your letter writers, and that these sort of things are likely to vastly outweigh anything you could write in your cover letter.
As someone who has chaired several search committees in the last five years, I can tell you that there's very little chance that I'd read a cover letter that was longer than 2 pages long. I've got hundreds of applications to read on mathjobs.org, so there just isn't time for this.
The cover paper can be helpful in explaining why you're particularly interested in our position (a spouse working in a nearby city, family who live in the region, interest in research collaboration with faculty in other departments.) It shouldn't take more than a couple of paragraphs to explain that kind of stuff.
You will have a separate CV, research statement, and teaching statement, so those aspects of your application don't belong in your cover letter.
What is your coach suggesting you put into this 2-3 page long cover letter?