Dear Chris & other colleagues:
Despite any advanced degrees, my advice is to "go small, or you may have to go & be 'stuck at home'."
In my view, small museums are the very best places to start a career so as to be given significant high-level responsibilities right off the bat. You then gain the widest kinds of important experience in all aspects of the field at small institutions where you might be the only employee--to say nothing about it being easier work to get.
You likely will need to move to a small community, but--depending on the local culture--there you can be a 'big fish in a small pond'. Local leadership roles beyond your institution will be tossed into your lap. In small institutions, you will be able to put all your training into practice & have wide latitude to make the changes you believe are the most important & necessary to increase the professionalisation of the operations flowing out of your education.
This early experience & demonstrable rapid results will pay the biggest dividends when looking to sell your EMP experience to advance a career in future. I would say, "Take a low-level job in a large institution only as a last resort."
Your search should prioritise municipally owned & operated institutions. My first F-T permanent job--where a $1.7 million CAD capital project was in the offing--as well as 3 others were at unionised municipal & other levels of government shops => good pay & benefits as well as secure funding vs. precarity.
Although, as we learned from one request for "help!" a couple or 3 years ago on this platform, it is not unknown for a local government to cut a museum. Later in my career while I was working as ED for one non-government local historical society, a single failed grant application in the face of an ongoing capital project resulted in the need for me to recommend to the board that they lay me off for some months in order to retain the job of my only lower ranked co-worker over the winter.
All the above is to say, a "tax base"--even one in a small municipality--is a really firm foundation for a career in museum work. You may even find you want to make an entire profession out of it!
I have never regretted starting small & have benefitted from it beyond measure.
Paul
P.S.: One caution for the above advice is the danger of burning yourself out as a 'lone director' of a small museum, but there are ways & means of avoiding the growing problem as recommended at
https://solvetasksaturation.wordpress.com/ .