You certainly get around. I've found that if you stay in one place, do a good job and agree terms before you start, then a mutually beneficial arrangement can be reached.
A teacher who stays in the same 'cozy' place for too long becomes complacent, lazy, and starved of inspiration . And before you ask, I say that from a lifetime of teaching experience, in both public and private sectors. When I hear of a teacher who retires after being in one school for most of their career, my sympathies go to all the students who were starved of inspirational teaching, and creativity... by a teacher who either so lacked confidence in their own abilities that they were afraid to face new challenges from time to time, or was just too lazy to!
We all learn and develop as individuals and teachers by expanding our experiences, not by adopting insular practices, which virtually guarantee that our methods and ideas will never be challenged. Complacency is a well recognised trait amongst some teachers. So perhaps a little less smugness and an appreciation of an alternative opinion to you own may just be worth considering.
As a senior manager, when appointing staff I am far less likely to be impressed by a candidate who does not have a range of different experiences, gained in at least several different institutions.