From my perspective at a regional university: most of our hires in math are new PhDs or have just a few years after PhD. But the application you need to be competitive for teaching-focused schools and positions is quite different than what you need for research-first positions.
For postdocs and tenure-track jobs at R-1 schools, you want to emphasize your research, while showing that your teaching is decent and not likely to cause complaints among the students. Bland teaching, to some degree, is a good thing - if you focus "too much" about teaching, it may cause people to worry about your research productivity.
For tenure-track jobs at teaching-oriented schools, you want to demonstrate that you will be excellent at teaching, not just unobjectionable. As much as possible, you want to demonstrate a history of teaching excellence as a graduate student (and after graduation, if applicable). If you focus "too much" on research, it may cause people to worry about your teaching quality.
Many schools "in the middle", including mine, are looking to increase their research profile, so we require much more in the area of research than we did 20 years ago. But we still look for teaching excellence, not just competence.
One common mistake (particularly among people who are sending out hundreds of applications) is sending the same application everywhere. If you send a research-focused application to a school where teaching is the primary criterion, you are not likely to make it past the first round of cuts. With hundreds of applications for the position, there will almost certainly be other candidates who have similar research accomplishments and demonstrated teaching excellence.
My advice for graduate students in general is to keep in mind the type of position you'd like to have 10 years after getting your PhD, and begin to groom your vita during graduate school to be competitive for that type of position. This may be easier said than done, of course.