For mid to longterm stays
Many years ago I was in a situation where I needed a place to stay for a few months. Regular hotels would be too expensive so I looked around and discovered monthly weekly mansions. These are apartments that allow you two rent by the week or the month and only require identification unlike regular apartments which require large deposits and gift money for the owner.
There are national chains that offer this service but I recommend finding a local establishment wherever you will be staying. They are likely to be less expensive and more flexible.