As I reflect on my academic career so far, I realize one could view it as a natural experiment on the question of whether scholarly productivity as such leads to multiple job offers. I am kind of the ideal test subject because I lack other obvious markers of prestige — my PhD is not from a top-tier school, and until recently, I taught at a place that was, shall we say, very little known. On the other hand, I don’t think it’s a boast to say that I am in the top 1% in sheer scholarly productivity among my age cohort in the humanities. So if publication volume, simply taken in itself, were a sure-fire ticket to multiple academic job offers, then I would be experiencing that. Hence I conclude that the answer is no.
This is not to say that it should. My publication record is a pretty abusive baseline expectation for a comfortable middle-class job, and if every aspiring academic published as much as I do, there would be an unimaginable glut of material. From my own perspective, I do in fact have a satisfying job at a great school. And I didn’t do all this work so that I could get a job — I did it because I enjoy it, and I have gotten the rewards (great interlocutors, invitations to come speak, etc.) that are really important to me from my work.
But for the young academics out there — no, sheer volume of publications is not a silver bullet. Write and publish as much as you want to and can, but don’t do it in the expectation that the academic job market will directly reward you for the length of your CV. And, I would say, you shouldn’t make serious sacrifices for the sake of writing projects you wouldn’t have taken on through your own sincere interest and passion, just for the sake of building your CV. That’s just not how it works. I don’t pretend to know how it does work, but I’m pretty sure at this point, I know better than anyone that it doesn’t work in this particular respect.
An und für sich 
My sense is that there’s something of a law of diminishing returns here. I went through enough years on the job market to be convinced that there’s *no* silver bullet whatsoever. Or, rather, for any given job, there’s about 72 silver bullets, all of which can be demanded at a moment’s notice, but whose precise constellation is entirely inscrutable from outside the search committee (and likely for the search committee itself).
I’m convinced that my (much more paltry) publication productivity contributed to my meager success on the job market (I did end up in a TT position after 5 years, but my “success” mostly consisted of pyrrhic victories—interview invitations are great and all, but….). I don’t know what the “line” would be, but I think that without a few book chapters and an article or two, I wouldn’t have gotten many of the interviews that I did.
All that to say, there’s no bar at which publication productivity will earn you a position, but there’s probably a bar under which you’re almost certain to be denied one (absent a prestige PhD., famous/powerful relatives, etc.).
Right, of course you should publish something. What I generally tell grad students is that they should teach a couple classes (if they haven’t already done so independently as TAs), just to show they can. Doing too many slows you down and makes you look like an adjunct dead-ender. Not sure how the parallel would work with overpublishing.