My health insurance was scheduled to end 1 month before my new postdoc started, but the postdoc is going to pick up health insurance for the two months prior to my starting to make sure that I am able to maintain continuity. (Their administration requires things come in 2 month blocks, so I'll be double-insured for one month.) Very nice of them, and beyond the call; they really did not have to do this. I'd even briefly shopped around for gap policies.
When I spoke with the postdoc program manager for the department, she told me all about the people in the area, and where I might want to live. She was conscious that postdocs are sometimes isolated because they work on only one project, and I'd have to work hard to put myself out there, but they'd put me on the student listserv and encouraged me to come to all the student events. I'm really excited to hear how conscious they are of everything.
By contrast, in my current position, the administrator for the research center within the larger research institute introduced me to maybe one person and was never conscious that I have a good experience.
I'm really impressed by them.
She also asked about the faculty position I'm waiting for, and I said I wasn't sure, but at this point it's so late that if I got the position I would try to start a year later. As much as I dread moving where I don't know anyone, and I know it will still be difficult no matter how nice people are, I'm so relieved at how welcoming they've been so far and that they are conscious of making sure that the transition goes smoothly.