Negotiating pay when I have been out of work (update)

I do not know whether to give a range or ask for theirs. I have read a lot of threads here and could not find my exact situation. I looked up ranges in Birmingham and they vary a lot. How do you research fairly?

Benefits matter too. A slightly lower rate with better coverage was better for me.

I gave a range based on local postings and said I was flexible depending on the whole package. They did not push back.