From the experience of family members I can tell you for many districts the key criteria beyond minimum qualifications are: 1.Who do they know in the town/district? and 2. will they take the lowest starting salary? I have been told off the record by a senior administrator that they are not interested in paying anything beyond the lowest possible starting salary because that is what the board wants. It's not stated anywhere of course. This is a top 30 public district in NJ.
You're forgetting "What subject do they teach?"
Years back when I was teaching, we had 5 open math positions at the end of one year.... and the district got 2 applicants by mid-August. Two more were filled by late-entry alternate route candidates (one of whom quit after winter break), and another by a long term sub (who was replaced by a BA teacher in October, who subsequently quit at the end of the year).
Math/science teachers are not easy to find... and they tend to have a higher attrition rate.
A friend graduated Montclair State with an education degree, with a focus in math. He went to a "getting hired" seminar just prior to graduation, where they talked about building portfolios, resumes, etc. Everything was about samples of student papers, sample writing prompts and assignments, etc. He raised his hand and asked if there was anything specific to math positions. The guy running the seminar then asked how many math educators were in the auditorium.... and my friend and one other person raised their hands. He told them to bring a mirror to the interview to prove they were breathing, and they'd do just fine.
On the flip side, we had 2 English positions open the one year, and there were something like 30 applicants.